Cosplaying comes in many forms; there are those that hunt through the Internet looking for the perfect pieces, others buy a costume off the rack and some hire someone else to make it. Then there are those who painstakingly make almost all of their cosplay themselves.
The methods and techniques are as varied as the costumes themselves. Even the materials used vary greatly.
One common material used by many is EVA foam. It is extremely versatile and not that expensive. This is a flexible, rubbery material that comes in various thicknesses and sheet sizes and is frequently used as floor mats.
If you are looking at building your own cosplay, a great place to start looking is at The Foam Cave. He has a great set of introductory tutorials that lay out the suggested tools, thicknesses and techniques.
If you are interested in creating your own cosplay, check out the Foam Cave