There are as many ways to operate a puppet as there are types of puppets, actually even more considering some puppet-makers build each puppet with its own unique method of operation.
But most standard rod, sock and human arm puppets are made to be operated pretty much the same way, with straps built in to the mouth plates to hold the fingers in while moving the mouth and a small mechanism used to move the eyes and/or eyelids.
And as you'll see in this WIRED video featuring puppeteers from Sesame Street, operating the seemingly simplistic puppets like Abby Cadabby can be rather difficult, while large puppets like Snuffleupagus are easier to control than you'd think.
-Via Geeks Are Sexy