A few questions.
Does your old bike have a threadless, or threaded headset?
What size steerer, 1" or 1-1/8""
Since you describe it as an older bike, I'll answer as if you have a 1" threaded headset. In that case the adapter fits into the fork and tightens with a wedge like your old quill stem. Then it has a roughly 2.5" 1-1/8" dia stub sticking out above the headset to which you can clamp the stem.
Here's a link to various models. Note the bottom has to fit the fork/headset, and the top has to match the stem.