It was cool seeing the mapping of the squares on the surface change as the character was moving - I was wondering if there is a reason the surface is composed of squares as opposed to something like triangles and also if there is a convention for which shape to use based on the type of surface.

This is kinda a hard one to answer...There are basically two types of modelling in play - one for game models and one for basically everything else, like movies. To grossly oversimplify, the difference is based on time v. quality and how subdivisions work - it is easier to subdivide a quad than a triangle while maintaining curvature. An easy way to see this in action is to try modelling 2 cubes, one subdivided with quads and another with triangles and note the difference.

I am wondering how we can create some models like this?