Several approaches attempt to classify projections. Most are orthogonal, thus any single projection may belong simultaneously to different categories. In others, like biologyinspired Maurer's, a branching taxonomy is applied.
Please note that, for ease of definition and visualization, some categories and projections are informally described below in terms of parallels and meridians, thus some properties may seem to depend on the particular aspect used for the map. For instance, the coordinate lines in cylindrical maps cross at right angles in equatorial, but not in polar or oblique maps, although all other properties still hold; after all, the coordinate grid is only a set of conventional lines.
Category  Properties  
Azimuthal  Also called zenithal. Shows true directions (azimuths) from a single point; in polar aspects all parallels are circular, and meridians are straight lines uniformly spaced and concurring at a point; an unclipped world map is a disc.  
Cylindrical  Defined by analogy to a cylinder as an intermediate projection surface; in the equatorial aspect all parallels and meridians are straight lines; meridians are orthogonal to parallels and uniformly spaced; an unclipped world map is rectangular  
Conic  By analogy to a conic intermediate projection surface; in the polar aspect all parallels are concentric arcs of circle, while meridians are straight lines perpendicular to every parallel, uniformly spaced by less than on Earth; unclipped maps are circular or annular sectors  
Pseudocylindrical  In the equatorial aspect all parallels are straight parallel lines; meridians are arbitrary curves, equally spaced along every parallel  
Pseudoconic  In the polar aspect all parallels are concentric circular arcs, while meridians are arbitrary curves  
Arbitrary or compromise  Parallels and meridians are arbitrary curves; usually no purely geometric construction is defined. Some authors call “arbitrary”, “conventional” or “compromise” any projection not derived from geometric devices, but customfit to a purpose 
In a sense, the cone includes as extreme cases both the cylinder (a cone with vertex at the infinite) and the plane (a cone with zero height). Therefore, the conic group generalizes the azimuthal and cylindrical and, broadly, pseudocylindrical and pseudoconic projections. Also, some consider a polyconic group to include projections where parallels are derived from circles, including modified azimuthals like Hammer's and Aitoff's. Actually, many socalled “azimuthal”, “conic” or “cylindrical” projections are not built on a pure projective process using solids, but are so classified due to geometrical properties of the mapped coordinate grid.
Also, a projective, geometric or perspective projection can be described in exact analogy to a geometric setup of light rays connecting the original surface to the map surface. Some authors call other projections “mathematical”.
Category  Properties  
Equalarea  Any region in the map has area directly proportional to the corresponding region on the sphere; also called equivalent or authalic. Generally more useful for statistical comparisons and didactic purposes.  
Equidistant  On the map there are two sets of points A and B, such that, along a selected set of lines (not necessarily straight), distances from any point in A to another in B are proportional to the distances between corresponding points on the sphere, again along those corresponding lines. In other words, scale is constant on those lines, which are called standard. Most projections have such sets but few are actually called “equidistant”.  
Conformal  In any* small region of the map, two concurrent lines
have the same angle as corresponding lines on the sphere,
thus shapes are locally preserved. Also called
orthomorphic or autogonal. Most
important for navigational purposes and largescale mapping,
especially in the ellipsoidal case.
*In almost every conformal projection, at least one point (frequently a pole is chosen) either can not be represented or fails conformality. 

Aphylactic  Some authors use this name for those projections which are neither conformal nor equivalent. 
Projections enumerated below are described in further detail. This is just a small sample of all existing designs, not necessarily the most important or most commonly used; as such, the selection is necessarily subjective. Equally arbitrary is whether changing the aspect or another minor detail is enough to justify a separate entry (cf. e.g., Cassini's v. equidistant cylindrical, Gauss transverse v. Mercator and Petermann v. Berghaus).
Sample graticule 
Common names
Main Features 
azimuthal orthographic, orthographic
Azimuthal, “realistic” view of Earth as seen from
space infinitely away. Shows at most a single hemisphere 

azimuthal stereographic, stereographic
Azimuthal, conformal, preserves all circles; shows at most
one hemisphere 

gnomonic, central, centrographic, gnomic
Azimuthal, all great circles map to straight lines;
extreme distortion far from the center; shows less than one
hemisphere 

general vertical perspective
Azimuthal, general case of
orthographic,
stereographic,
gnomonic
azimuthal projections. Most realistic view from space directly
towards center of Earth. Parameterized by distance of projection
center; special cases by La Hire, Parent, Lowry, Fischer,
Gretschel, James, Clarke (“Twilight”) and others


azimuthal equidistant, zenithal equidistant
Nonperspective azimuthal,
preserves distances along any line
touching the center of the map 

Lambert's azimuthal equalarea
Nonperspective. Unique azimuthal equalarea projection 

Ginzburg's azimuthal I and II
Nonperspective, neither equalarea nor conformal 

Lambert's equalarea cylindrical; variations by Behrmann,
Trystan Edwards, Gall's (orthographic), Peters, Dyer, Tobler/Chen
Only possible cylindrical equalarea projection, including
scaled variants like Gall's (“Peters”) and HoboDyer which differ only in
standard parallels


Gall's stereographic cylindrical
Neither conformal nor equalarea. Variations include
Braun's stereographic
cylindrical and the BSAM cylindrical


Braun's stereographic cylindrical
Neither conformal nor equalarea, special case of
Gall's stereographic


central cylindrical, centrographic cylindrical
Neither conformal nor equalarea; not to be confused
with Mercator's.
Transverse aspect is the Wetch projection


equirrectangular, equidistant cylindrical, plain chart, plane chart;
special cases are the simple cylindrical (plate carrée),
Gall's isographic and Cassini
Cylindrical, true scale along all meridians and one or two parallels; neither conformal nor
equalarea; in the most common case maps into a rectangle with aspect
ratio 2 : 1 

Cassini
Transverse aspect of the plate carrée 

Gall's isographic
Special case of the equidistant cylindrical, standard parallels
45°N and 45°S 

Mercator, cylindrical conformal
; transverse ellipsoidal form called
Gauss conformal or GaussKrüger
Only possible conformal cylindrical projection;
transverse aspect is the foundation of the
UTM grid 

Miller
Cylindrical, arbitrary compromise to
Mercator;
neither equalarea nor conformal


Trapezoidal, Donis
Pseudocylindrical, meridians are straight lines,
sometimes symmetrically broken at the Equator 

Mollweide, elliptical, Babinet, homolographic, homalographic
Pseudocylindrical, equalarea, meridians are
ellipses; full map bounded by 2 : 1 ellipse;
sometimes interrupted; variations
include Atlantis and Bromley's


SansonFlamsteed, sinusoidal, Mercator equalarea
Pseudocylindrical, equalarea, meridians are sinusoids,
parallels are equally spaced and standard lines; 2 : 1 

Foucault's stereographic equivalent
Pseudocylindrical, equalarea, parallels are spaced like in the equatorial
aspect of the azimuthal stereographic 

Collignon
Pseudocylindrical, equalarea, meridians are straight
lines. Two main variants, with triangular frame or
symmetrical diamond with meridians broken at Equator 

Craster parabolic
Pseudocylindrical, equalarea, meridians are parabolas. Same
as Putniņš's P_{4} 

Loximuthal
Pseudocylindrical, all straight lines passing through intersection of
central meridian and a reference parallel are loxodromes with correct
scale and azimuth. Usually asymmetrical around the Equator
 
Quartic authalic
Pseudocylindrical, equalarea, meridians are 4th order
polynomials; limiting case of
Hammer
and
EckertGreifendorff 

Flat polar quartic
Pseudocylindrical, equalarea, poles are 1/3 as long as the Equator 

Nell's pseudocylindrical
Pseudocylindrical, equalarea, polelines 

NellHammer
Pseudocylindrical, equalarea, polelines


Eckert I
Pseudocylindrical, 2 : 1, poles
are half as long as the Equator, meridians are straight
lines broken at Equator. Parallels equally spaced


Eckert II
Pseudocylindrical, equalarea, 2 : 1, poles
are half as long as the Equator, meridians are straight
lines broken at Equator


Eckert III
Pseudocylindrical, 2 : 1,
meridians are elliptical arcs (boundary is circular).
Parallels are equally spaced 

Eckert IV
Pseudocylindrical, equalarea, 2 : 1,
meridians are elliptical arcs, circular at boundary


Eckert V
Pseudocylindrical, 2 : 1,
meridians are sinusoids, parallels are equally spaced.
Particular case of Winkel's first projection


Eckert VI
Pseudocylindrical, equalarea, 2 : 1, poles
are half as long as the Equator, meridians are sinusoids


Rosén's pseudocylindrical
Pseudocylindrical, equalarea, based on
sinusoidal: poles
are mapped to parallels arcsin(0.8) N and S of base projection


Robinson, orthophanic
Pseudocylindrical, compromise. Neither conformal nor equalarea
 
Kavrayskiy V
Pseudocylindrical, equalarea
 
Kavrayskiy VII
Pseudocylindrical, compromise, elliptical meridians
 
Goode homolosine
Pseudocylindrical, equalarea, hybrid joining
Mollweide
at poles, SansonFlamsteed
at the equatorial band, almost
always interrupted 

Boggs eumorphic
Pseudocylindrical, equalarea, arithmetic average of
Mollweide
and SansonFlamsteed
projections. Usually interrupted 

SinuMollweide
Pseudocylindrical, equalarea, hybrid fusion of
Mollweide
and (in lower portion) SansonFlamsteed
projections. Usually oblique and interrupted 

Winkel I
Pseudocylindrical (generalizes
Eckert V),
averages SansonFlamsteed and
equidistant cylindrical,
meridians are sinusoids 

Winkel II
Pseudocylindrical, averages
equidistant
cylindrical and a modified elliptical
projection 

HEALPix
Pseudocylindrical, equalarea, hybrid of Lambert's
equalarea cylindrical and interrupted Collignon's projection;
designed for raster processing of astronomical and cosmological data in
the FITS grid


PseudoEckert
Pseudocylindrical, equalarea, meridians are
partial sinusoids 

Perspective (orthographic, stereographic or centrographic) conic
Conic, true perspective. Used by Murdoch and Colles


Equidistant conic
Conic, constant meridian scale; limiting cases are
azimuthal
equidistant and
cylindrical
equidistant projections. General case of Schjerning's I
projection. Many variations, mostly in
choice of standard parallels (Murdock, Euler); others
include de l'Isle's coniclike projection


Braun's stereographic conic
Perspective conic with center of projection at a pole and 30° standard
latitude


Albers equalarea conic
Conic, equalarea; limiting cases are Lambert's equalarea
conic and cylindrical
projections 

Lambert's equalarea conic, isospherical stenoteric
Conic, equalarea; limiting case of
Albers's
conic, with a pole as standard parallel


Lambert's conformal conic, orthomorphic conic
Conic, conformal; limiting cases are
azimuthal stereographic
and
Mercator projections


Polyconic, American Polyconic
Polyconic, parallels are nonconcentric arcs of circle with
correct scale. Neither conformal nor equalarea. 

Rectangular Polyconic, War Office
Polyconic, parallels are nonconcentric circular arcs crossing
all meridians at right angles; either the Equator or two parallels
have correct length. Neither equalarea nor conformal 

Wiechel
Pseudoazimuthal; modified
azimuthal equalarea
projection, no longer azimuthal. Only interesting in polar aspect,
where meridians are circular arcs with standard scale. Usually clipped
to a single hemisphere 

Aitoff
Stretchhing of modified
equatorial
azimuthal equidistant
map; boundary is 2 : 1 ellipse; neither equalarea nor conformal 

Hammer, HammerAitoff, AitoffHammer
Modified from
azimuthal
equalarea equatorial map;
equalarea, boundary is 2 : 1 ellipse; variations
include Briesemeister,
oceanic
and Nordic 

Briesemeister
Rescaled oblique Hammer projection.
Equalarea 

EckertGreifendorff
Similar to Hammer projection, with
doubled rescaling factor and therefore almost straight parallels.
Equalarea 

Schjerning II
(original map includes an arbitrary
unspecified enlargement)
Azimuthal equidistant
inner hemisphere. The outer hemisphere completes a 2:1 ellipse.
Distances correct from center of map, azimuth correct only in
inner hemisphere. Not conformal or equalarea 

Schjerning III
Map comprises two circles joined at a point. Distance from the center of
the map to every point is correct, but not azimuth. Final map centered on
London. Neither conformal nor equalarea 

Wagner IX, AitoffWagner
Modified
Aitoff projection;
neither equalarea nor conformal


Winkel Tripel
Arithmetic mean of Aitoff and
equidistant
cylindrical projections. Neither equalarea nor conformal 

StabiusWerner I
Pseudoconic, equalarea, parallels are equally spaced
circular arcs centered on a pole


Werner, StabiusWerner II, cordiform
Pseudoconic, equalarea, parallels are equally spaced
circular arcs and standard lines, centered on a pole.
Schjerning IV
is an oblique aspect;
Schjerning V
has shortened parallels,
Schjerning VI is interrupted


StabiusWerner III
Pseudoconic, equalarea, parallels are equally spaced
circular arcs centered on a pole


“Bonne”
Pseudoconic, equalarea, parallels are equally spaced
circular arcs and standard lines. Appearance depends on reference parallel.
General case of both
Werner and
sinusoidal


“Lagrange”
Meridians and parallels are circular arcs, except the central meridian and
a base parallel which are straight.
Conformal except at the poles. The basic case, developed by Lambert, is
circular 

DeLucia/Snyder's orthographic projection of Gilbert's conformal double
world sphere
Graticule comprises elliptical arcs. Neither conformal nor equalarea.
 
Peirce Quincuncial


Guyou


Adams's hemispheres on squares


Adams's world on a square (1929)
Poles at opposite vertices; Equator along a diagonal.
Conformal except at four vertices 

Adams's world on a square (1936)
Poles at midpoints of opposite edges. Conformal except at poles and
four vertices 

Xarax's world in half a hexagon
Threelobed rearrangement of Lee's map on a tetrahedron.
Conformal except at midpoints of three longest edges 

Eisenlohr
Fully conformal, no singular points. Scale constant along
boundary. Optimal range of scale distortion for a conformal design 

August, August epicycloidal
Conformal everywhere, with no singular points. Map bounded
by a epicycloid. Base for some of Spilhaus's
oceanic maps 

Van der Grinten, Van der Grinten I
Boundary is a circle, meridians and parallels are
circular arcs, except central meridian and Equator.
Not conformal, large area distortion far from Equator 

Van der Grinten II
Boundary is a circle, meridians and parallels are circular arcs
intersecting at right angles; straight central meridian and Equator.
Neither equalarea nor conformal 

Van der Grinten III
Boundary is a circle, meridians are circular arcs; straight
horizontal parallels intersect straight central meridian at the same
points as in Van der Grinten I.
Not conformal or equalarea 

Van der Grinten IV
Bounded by two intersecting circles, meridians are arcs
of circle equally spaced along Equator, parallels are arcs of circle.
Neither conformal nor equalarea 

Bacon's globular
Single hemisphere bounded by a circle. Circular meridians; straight
horizontal parallels equidistant along hemisphere boundaries. Neither
conformal nor equalarea


Apian's first globular
Single hemisphere bounded by a circle. Circular meridians; straight
horizontal parallels equidistant along central meridian. Neither conformal nor
equalarea. Extended by Ortelius and Agnese


Apian's second globular
Single hemisphere bounded by a circle. Elliptical meridians; straight
horizontal parallels equidistant along central meridian. Neither conformal
nor equalarea


Fournier's first globular
Single hemisphere bounded by a circle. Elliptical meridians; circular
parallels. Neither conformal nor equalarea


Fournier's second globular
Single hemisphere bounded by a circle. Elliptical meridians;
straight parallels. Neither conformal nor equalarea


Globular, “Nicolosi” globular
Single hemisphere bounded by a circle. Circular parallels and
meridians. Neither conformal nor equalarea. Also attributed to La Hire and
alBiruni


Ortelius's oval
Simple extension of Apian's first globular hemisphere. Neither
pseudocylindrical, equalarea nor conformal.


Leonardo da Vinci's octant map
Octant map, bound by circular arcs; graticule uncertain, probably
neither conformal nor equalarea


Jäger star
Graticule comprising only straight lines. Eight unequal
lobes, each symmetrical in core and outer hemisphere. Parallels
linearly spaced in each lobe. Neither conformal nor equalarea


Petermann star
Parallels are concentric, equally spaced arcs of circle,
meridians are straight lines (most broken at the Equator). Neither
conformal nor equalarea. Sometimes described with unequal lobes


Berghaus star
Fivelobed version of
Petermann's projection


Maurer's S233
Graticule comprises straight lines, with constant spacing.
Neither conformal nor equalarea. Symmetrical case of
Jäger's projection


Maurer's S231
Parallels are concentric arcs of circle; central hemisphere is
a Lambert
azimuthal map. Lobe meridians are curved. Equalarea 

WilliamOlsson
Combines
Lambert azimuthal
projection in partial inner hemisphere with lobes using rescaled
Werner projection.
Equalarea 

Bartholomew's “Tetrahedral”
Core is a partial
azimuthal
equidistant hemisphere. Lobes are modified
Werner maps
with expanded parallel scaling. Neither conformal, equalarea nor polyhedral


“Flowerpetal”, Daisy
Transverse Mercator
lobes; central core uses an
azimuthal
equalarea projection. Part of the USGS's ISIS package


Conoalactic
Very similar to
Berghaus,
but the central hemisphere is based on equidistant conic;
not to be confused with Cahill's “butterfly” map 

Armadillo (orthoapsidal on torus)
Intermediate projection surface is a torus with radii 1 and 1; final
map is projected orthographically; neither equalarea nor conformal 

Orthoapsidal on ellipsoid
Intermediate projection surface is an ellipsoid; poles may be points
or lines, and meridians may optionally have constant scale;
the final map is projected orthographically 

ArdenClose
Arithmetical mean of
equalarea
cylindrical map and its transverse aspect; neither equalarea
nor conformal 

Tobler's projection for local maps
Fast rendering for small regions. Neither equalarea nor conformal.
Parameterized by reference parallel


Gringorten's projection
Equalarea on a square


Lee's map on a regular tetrahedron
Conformal except at tetrahedral vertices 

COBE Quadrilaterized Spherical Cube
Cubic, approximately equalarea. Intended for cosmological charts of the
universe, not earthly geography


Quadrilaterized Spherical Cube
Cubic, equalarea. Modification of COBE QSC 

Fuller's Dymaxion™ projection on a cuboctahedron
Scale preserved along face edges. Neither equalarea nor conformal.
Several possible face arrangements 

Fuller's Dymaxion™ AirOcean World Map on an icosahedron
Scale preserved along face edges. Neither equalarea nor conformal.
Several possible face arrangements 

Gnomonic projection on polyhedra; adopted by several
authors, notably Irving Fisher on the icosahedron
and Cahill on octahedra
Identical to ordinary gnomonic, with
advantages and shortcomings of interruptions, plus arbitrary
face arrangements


Fisher's equalarea projection on the icosahedron.
Equalarea. Generalized by Snyder for other regular polyhedra


Cahill's Butterfly
Developed on the regular (usually truncated) octahedron. Basic projection
is gnomonic; variants are equalarea or conformal. Further refined by
Gene Keyes


Steve Waterman's projection system
Based on a truncated octahedron defined by the centers of packed spheres.
Graticule comprises broken straight lines. Neither conformal nor equalarea


Kent Halstead's equidistant projection
Equidistant along all meridians and parallels, which are
broken and interrupted to reduce shearing.
Neither conformal nor equalarea


Kent Halstead's Composite World projection
Interrupted, based on Lambert's azimuthal projection. Mostly
equalarea, except at lobe boundaries
