In Torqueless' picture the modern A-Headset is on the left and the older type on the right.
The older type for use with quill stems require a threaded steerer for the top and locknuts to screw onto. The stem is secured in the steerer.
A-Headsets (or threadless headsets) are for use with threadless steerers. The stem clamps around the top of the steerer.
You can buy converters to use a modern stem with a threaded headset and steerer: it's basically a quill stem but without the horizontal extension. Personally, I don't rate them as they have none of the advantages, but all the disadvantages, of both systems.