There are many ways of making a stop motion puppet that you can animate.
The simplest is a blob of plasticine - that's oil based modelling clay that doesn't dry out. You can morph clay from one thing to another, and make any facial expression, but the downside is that you are constantly re-sculpting and smoothing it. With just the clay, you need to stick to fairly squat shapes that slide around.
For something more like a human or animal shape, you want an armature inside - that's a skeleton that can hold a pose. If it is going to walk naturally, it needs Tiedowns - something that holds the foot to the set floor so the puppet doesn't fall over, even when it's off balance.
The simplest and cheapest armatures are made from aluminium armature wire. It has been annealed so it is soft and can stand up to a lot of bending. I use it 99% of the time. It's a good place to start. Some art and craft/sculpting stores sell it, but not in every town so you may need to buy it online. One place in the US is whimsie - you want the Soft Aluminum Wire - http://www.whimsie.com/aluminum%20craft%20wire.html?gclid=CJSi4bjR6...
You can use flexible materials like cushion foam and liquid latex to build up a body over the wire armature. It is more permanent, and doesn't need re-sculpting each frame like clay does. There are a few tutorials on making puppets at Youtube. I have one on Tiedowns ( http://www.youtube.com/watch?v=jK1tAh_kCZE&feature=relmfu ), one on a build-up body ( http://www.youtube.com/watch?v=MbF6m3BeGUQ ), another on a similar build-up head ( http://www.youtube.com/watch?v=4uG-ZIOXZ7w&feature=relmfu ), and one on casting a silicone head for when you are more advanced.
Stop Motion Magazine has several, starting with this simple puppet ( http://www.youtube.com/watch?v=53iwFJrATmE&feature=related ).
Bluworm has some on how he makes his creatures ( http://www.youtube.com/watch?v=XniNAc1mfw0&feature=related ).
I've made my own plasticine, actually it's a play-doh but the problem was it dried during the shooting on some places so I had to re-sculpt some parts. Is it a play-doh (play dough) a good material to begin with and substitute clay since it's cheap to make and do you have any advice on how to prevent it from drying out so fast?
Thanks for the answer
No. It is better to use oil-based plasticine that does not dry out.
But, one way to make some materials that do not dry out quickly is to use glycerin instead of water. I don't know how well this would work for play-doh, but it might be worth doing a small test.
I will look for some of that oil based plasticine in art shops, although this homemade play-doh was a cheaper deal :D
Hm nice idea..I will try something like that, maybe it will work.