Professor Quirrell may be the Defense Against the Dark Arts teacher…but he's also a total wet noodle.
You spend most of this movie wondering what the heck he's doing teaching Defense Against the Dark Arts, since it looks like he could barely take on a little old lady at a rummage sale. But for some reason, Dumbledore trusts him and lets the stammering wimp teach the kids how to stand up and face the forces of darkness.
As it turns out, he knows a lot more about said forces of darkness than he's letting on… mostly because he's keeping Mr. Maximum Evil (a.k.a. Voldemort) slapped onto the back of his skull like a demonic tattoo.
The whole wet noodle thing was an act designed to throw off suspicion:
QUIRRELL: Next to me, who would suspect p-p-poor s-stammering Professor Quirrell?
Quirrell exists solely as a mobile home for Voldemort's spirit, and as such he wants to get his mitts on the Sorcerer's Stone awful bad. That ultimately makes him the villain of the piece, though we don't know it until the very end and his reward for being so is to turn to ash like a Cuban cigar. A fairly ignominious death, but then again, he really wasn't much of a villain. He was just possessed by one.