There are many different things that you can use to build a puppet. A sock, a spoon, fleece, a box- the list goes on and on.
My monstery puppet will be built using a skeleton I construct from foam. The poly foam is 1" thick, and sold in rolls at craft stores for use in cushions and such things.  I just picked up a new roll today- I try to watch for 40% off coupons for my local craft stores. The glue I use is called Weldwood contact cement. 
I have built many different types of eyes. You can make them from styrofoam balls, ping pong balls, plastic spoons, ping pong balls and many other things.  The eyes I will be using for this puppet, I order from a supply store. *We'll talk more about the eyes when I am on that step :)
I end up hand sewing quite a bit of the puppet. I am pretty rough on thread, and found that a spool of upholstery thread works the best for me. It makes things sturdier, and if I need to use a lighter weight thread in some areas I can always do that.

After the character sketch has been approved, it's time to choose the proper fabric. In the case of this monstery hand puppet, the furrier the better. After going to a few of my favorite online* fabric stores, I narrow it down to a couple different long pile furs.
Although the top choice is a nice long pile purple fur, I like the bottom fur better for it's wild "hairy" texture.

(*If you live in a large city- head to the fabric district! I am in the middle of Nowheresville, an hour from anywhere of significance.)

To build a puppet, I first have to get the character in my head down onto the paper. Sometimes this comes easily, other times there's kicking and screaming. Draw, draw, draw. Lots of paper might end up in the recycle bin, but better than fur and feathers.
I like to use Prismacolor markers and a Sharpie pen on plain printer paper. Once I get a black and white sketch that I am happy with, I scan it, then print it out. That way, if I mess up in the coloring process- I just print out another page and color again.