Essentially: conformal (angles are equal in different areas of the map -- this is the purpose of Mercator, for sailing), equal-area (regions of the same area are represented equally), contiguous (paths between regions do not leave the map, e.g. not a Goode homolosine, Dymaxion, etc). Pick two.

There is a lot of theory surrounding map projections. I only took an introductory GIS course in college, but it was clear that there is an incredible amount of work behind the scenes there.

http://en.wikipedia.org/wiki/Geographic_information_system

A spherical globe still counts as a projection of our non-spherical planet!

I've always preferred using a globe to get a perspective on the Earth and the various positions of continents and nations.