What different types of map projection systems are there?

Map Projections:

For a globe to be projected onto a two-dimensional surface, the plane is usually folded or wrapped into a three-dimensional shape so it can receive the projected image. The shape of the plane determines the resulting type of projection. The most conventional projections methods are Mercator, conic, and gnomonic projections: Mercator projections wrap the plane into a cylinder, conic projections wrap the plane into a cone, and gnomonic projections leave the plane flat.

"There is no one map projection which is better than all the others; a projection which is good for Asia is not necessarily the best for Africa, and a different projection is used for a navigation chart than for a statistical map. Each projection has certain virtues and certain limitations." - Raisz, 166.