The part of a bike that connects the fork to the handlebars, and the whole thing to the bike's bike frame. Headsets come in threaded and threadless variations, threadless being the preferrable type. I can't tell you anything about threadless headsets, except that they need wrenches, but threadless headsets work like this:
The headtube of the frame is the tube at the very front of the frame. Attached to the headtube is the headset, two doughnut shaped protrusions, one at the top and one at the bottom, that contain bearings. These bearings allow for smooth steering. Through the headset goes the steerer tube of the fork, which protrudes about 5cm from the top of the headtube. Clamped to this is the stem, which then attaches to the handlebars. To keep the whole thing together a top cap is attached to the top of the steerer tube via an allen bolt.
If you don't understand this explanation look at the front end of your bike, or go to a bike shop and ask them to explain. If you want to buy a headset I would recommend a Chris King or Cane Creek headset. | 
