Cartography is the science of mapmaking. It comprises many problems and techniques, including:
South America in selected projections at identical scale. Which projection is best? Which is right? The short answer is none, at least not all the time. Even if a single projection is used, just switching the aspect can also radically reshape the continents. 
One important concern of cartography is solving how to project, i.e. transfer points from an almost spherical lump of rock (our Earth) onto flat surfaces, either paper pages or computer screens.
Here are informally described important cartographic concepts, how maps are drawn and why there are so many different kinds of projections for world maps. You may start reading here and follow the buttons, or use this table of contents:
Introduction  A gentle introduction to tinkering with maps  
Basic definitions and concepts about the Earth, maps and the mapmaker's choices  
Fitting Map to Purpose  Useful map properties:


Mathematics of Cartography  How projections are created, including equations for:  
Main Projection Groups  Azimuthal projections, perspective or not  
Cylindrical projections, arbitrary or perspective  
Pseudocylindrical projections, pure, continued or crossbred  
Conic projections, nonperspective and polyconic  
Pseudoconic projections  
Modified azimuthal projections  
Conformal projections  
Other interesting projections  
Coping with Distortion 
Tilted and crooked projections: oblique maps  
Tearing Earth's skin: interrupted maps


Rebuilding the Earth into an exotic planet: polyhedral maps


Pieces of History  Projections developed before the Modern Age  
Projections at Work  Projections which helped making the world smaller: the Mercator and azimuthal equidistant  
Some unusual applications of map projections (in construction)  
Conclusion  Summary and table of depicted projections  
Resources and links  
Quick answers and personal rants 