Remove a Trapped Sink Stopper

If the sink stopper (also called a pop-up) has stuck in closed position, try the following.

Steps

Make sure stopper is dry and sink is empty of water. (See below for a tip if you have water in your sink)
Unscrew the stopper from the mechanism holding it by pushing it around with your fingers.
Attempt to maneuver from inside the plughole to change the height.
You will now be able to use the sink, albeit without a plug.

If your sink is full of water, empty it first by bailing it out with a plastic or paper cup. If it is a bathroom sink, you can empty the water right into the overflow-drain at the top of the sink if you have one, so you won't have to empty the water into another sink altogether.
There is an excellent diagram of the pop-up, clevis strap, pivot rod, and spring clip at Popular Mechanics.