Why would turning off a sink faucet stop the shower?
The part of the mixing valve that keeps the temperature/pressure in check has failed or clogged. So when you reduce the water flow to the show by opening a sink tap, the valve reacts to keep the temperature/pressure constant. Then when the tap is closed, the valve again attempts to react, but since the valve is broken, it does not react correctly.

The good news is this is easy to fix. Take the handle and the trim cover off. If your shower valve was installed correctly, you should be able to turn off the water to the valve by turning the two screws to either side of the valve. If not, you'll need to turn off the water to the whole house.

Removing the valve (sometimes called a cartridge) varies. Some have a threaded cap nut on the front that you just twist off. Others have a circular retaining clip around the outside of the valve assembly or inside the front of the valve. Once you've freed the cartridge, take it to a plumbing supply store and they can give you a replacement along with any additional steps you need to follow, such as lubricating or replacing O-rings.

Assemble in reverse, but put a fat line of plumber's putty behind the trim cover along the inside edge (leave a 2 inch gap long the bottom).

Here's a good set of instructions with pictures: http://www.homedepot.com/webapp/catalog/servlet/ContentView?pn=KH_PG_BA_Repairing_Cartridge_Tub_Shower_Faucet&amp;storeId=10051&amp;langId=-1&amp;catalogId=10053