Grab some liquid drain opener from your local convenience store. I have no brand loyalty here - they are all basically the same.
Caustic drain cleaners are good for home use. They contain either sodium hydroxide (lye) or potassium hydroxide (caustic potash). This stuff loves to dissolve proteins in clogs.
Dump around two cups of the stuff down your drain. Depending on how bad the clog is, it might just run into the drain or sit in the sink.
Let it sit for 15 minutes.
So what is going on here? The chemicals in the drain cleaner are now reacting with the proteins in the clog (typically hair). This should turn the clog in to a mass that can be dissolved by water.
Flush the drain with hot tap water. If the drain still appears to be clogged, repeat the process.