What Material Chooseth You?

Often, choosing the material for your prop can be the most difficult part of the process; it will in fact determine the process. Choosing the wrong material can lead to added expense, additional labor and a whole lot of headaches. It can even result in a prop that does not look or perform as it should, with the only way to fix it being to rebuild it from scratch.
How do you know which materials to build your prop out of?