Historically cups have not been used.  For spaceflight, liquids have been stored in sealed bag-like containers and then drunk through a tube.  These are somewhat similar to how many children's drinks are packaged and stored.  This [document][1] shows some pictures of these sorts of containers and some food containers.

As [another answe][2]r has said, there have been demonstrations of drinking containers that take advantage of certain physical properties which allow a liquid to stay inside an open container.

There is also [video][3] of an astronaut letting water loose inside his cabin and then taking it into his mouth by moving his head into a position which allows that.  This was part of an educational video in which astronauts were showing how liquid behaves in a low gravity environment.

  [1]: http://ftcsc.ag.iastate.edu/media/package.pdf
  [2]: https://space.stackexchange.com/a/139/36
  [3]: http://www.youtube.com/watch?v=4o1lr9LxODc