Remove the pop-up plug for your sink by twisting and pulling up or by removing the pivot rod on the outlet pipe. Clean out any hair that is clogging the drain. Use your fingers or needle-nose pliers to grab hold of the hair in the drain and pull out.
Place a sink plunger over the drain, completely covering the opening. Cover the overflow hole with your hand or a washcloth. Push down on the plunger and then pull up hard on the handle. Do this until the clog starts to break and the water drains.
Place a bucket under the P-trap under the sink; remove the trap using a pipe wrench and loosening the nuts holding the trap in place. If the trap is clogged, clean it out and rinse and remaining gunk out of the pipe in another sink. Replace the trap and tighten the pipe nuts with the wrench.
Use a small hand auger to clear the drain if all else fails. You can buy an auger at your local hardware store. Pull a foot of cable out and insert it into the drain until it stops. Turn the auger clockwise until it starts to go into the drain further. Continue turning the auger until it breaks through the clog and the sink drains. Rewind and clean before putting it away.
Pour 1/2 cup of baking soda into the drain and then follow with 1 cup of vinegar. Let the combination combine and fizz for about 5 minutes to clean out the scum from the drain. Rinse with hot water.