The world according to Lambert's azimuthal equalarea map, in the equatorial aspect. Unprojected data by NASA 
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 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 and Gauss transverse v. Mercator).
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 a 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, distances from center are preserved, navigational purposes 

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 (orthographic), Peters, Dyer, Tobler/Chen  Only possible cylindrical equalarea projection, including scaled variants like Gall's and HoboDyer 

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 simple cylindrical (plate carrée), Gall's isographic and Cassini  Cylindrical, very fast and easy to compute, neither conformal nor equalarea; in the most common case maps into a rectangle with aspect ratio 2 : 1 (twice wide as tall) 

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 

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

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 P_{4} 

Loximuthal  Pseudocylindrical, all straight lines passing through intersection of central meridian and a reference parallel are loxodromes with correct scale and azimuth 

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 Equator or two parallels have correct length. Neither equivalent 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  Stretching of modified equatorial azimuthal equidistant map; boundary is 2 : 1 ellipse 

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

Briesemeister  Equalarea, simple oblique stretching of Hammer projection 

EckertGreifendorff  Rescaled modification of Hammer projection. Equalarea 

Schjerning II (original map includes an arbitrary 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 every point to the center of the map is correct, but not azimuth. Final map is 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 

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 double world sphere  Graticule comprises elliptical arcs. Neither conformal nor equalarea. 

Peirce Quincuncial  World map in a square, central hemisphere in an inner square. Conformal except at edge midpoints. Other aspects by Guyou and Adams 

Guyou  World map in 2:1 rectangle. Conformal except at hemisphere corners. Other aspects by Peirce and Adams 

Adams's hemispheres on squares  Hemispheres in two squares. Conformal except at square corners. Other aspects by Guyou and Peirce 

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

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

Xarax's world in half a hexagon  Threelobed world map derived from Lee's map on a tetrahedron. Conformal except at North pole and meridian breaks at each lobe 

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 Spilhaus's oceanic map 

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

Van der Grinten II  Boundary is a circle, meridians and parallels are circular arcs intersecting at right angles, except central meridian and Equator 

Van der Grinten III  Boundary is a circle, meridians (except central) are circular arcs; parallels are horizontal lines intersecting central meridian at same points as in Van der Grinten I 

Van der Grinten IV  Bounded by two intersecting circles, meridians are arcs of circle equally spaced along Equator, parallels are arcs of circle 

Maurer's fullglobular  Meridians along lines of Van der Grinten's IV, outer meridians bounded by half limiting circles. Parallels are arcs of circle, equally spaced both on outer meridians and Equator. 
Bacon's globular  Bounded by a single hemisphere in a circle. Circular meridians; straight horizontal parallels equidistant along hemisphere boundaries. Neither conformal nor equalarea.  
Apian's first globular  Bounded by a single hemisphere in a circle. Circular meridians; straight horizontal parallels equidistant along central meridian. Neither conformal nor equalarea. Extended by Ortelius and Agnese.  
Apian's second globular  Bounded by a single hemisphere in a circle. Elliptical meridians; straight horizontal parallels equidistant along central meridian. Neither conformal nor equalarea.  
Fournier's first globular  Bounded by a single hemisphere in a circle. Elliptical meridians; circular parallels. Neither conformal nor equalarea.  
Fournier's second globular  Bounded by a single hemisphere in a circle. Elliptical meridians; straight parallels. Neither conformal nor equalarea.  
Globular, "Nicolosi" globular  Bounded by a single hemisphere in a circle. Circular parallels and meridians. Neither conformal nor equalarea. Also attributed to La Hire and alBiruni.  

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 

Conoalactic  Very similar to Berghaus, but northern hemisphere is based on equidistant conic; not to be confused with Cahill's "butterfly" map 

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

Maurer's S231 (equalarea star)  Parallels are concentric arcs of circle; northern hemisphere is a Lambert azimuthal map. Southern meridians are curved. Equalarea 

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

Bartholomew's "Tetrahedral"  Central core is part of azimuthal equidistant hemisphere. Outer lobes are modified Werner 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. 

Leonardo da Vinci's octant map  Octant map, bound by circular arcs. Neither conformal nor equalarea 
Armadillo (orthoapsidal on torus)  Intermediate projection surface is a torus with radii 1 and 1; final map is projected orthographically  
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  For small regions only. Neither equalarea nor conformal. Parameterized by reference parallel.  
Gringorten's projection  Equalarea on a square.  
Trapezoidal, Donis  Pseudocylindrical, meridians are straight lines, sometimes symmetrically broken at the Equator  

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 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  Identical to ordinary gnomonic, with advantages and shortcomings of interruptions.  
Fisher's equalarea projection on the icosahedron.  Equalarea. Generalized by Snyder for other regular polyhedra  
Steve Waterman's projection system  Based on a truncated octahedron defined by the centers of packed spheres. 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. 