How can I fix a sink drain stop which won't hold water?
Sounds like a bad O-Ring on the stopper.  

Here is a good picture for reference.



To remove the stopper, you'll have to remove the retaining nut at the back of the drain body.  Once that's removed you'll pull the pivot rod out, this will allow you to remove the stopper.

Now that you have the stopper out, visually inspect the rubber gasket around the bottom of the plug (just under the chrome part).  If the gasket is worn or cracked, you'll have to either replace it, or the entire stopper.

If you take the stopper with you to the hardware store, somebody there should be able to help you find what you need.

To put it all back together, put the stopper in the drain, insert the pivot rod (so it interacts with the stopper), then tighten the retaining nut back down.  You can make minor adjustments by loosening the set screw that holds the lift rod to the clevis, and adjusting how deep the lift rod goes into the clevis.

WARNING:

Don't over tighten the retaining nut, or you could find that you won't be able to move the lift rod.