How to Stop a Dripping Sink
For about $3, you can easily fix most drips yourself. By far the most common type of faucet is the stem design. If your faucets have separate hot and cold knobs, they most likely are this type. First shut off the water under the sink. Turn the handle to open the faucet as if you were turning on the water. Use a wrench to remove the stem assembly. Take it along to the hardware store so you get the proper size washer. There are many similar sizes. Install the new washer in the bottom of the stem (usually held on by a small screw) and screw the stem assembly back into the faucet body.
Tighten with a wrench. Before tightening, make sure that the handle is turned as if the faucet were in the open position. If it is turned in the closed position with the washer and stem down, the washer may hit the valve seat before the stem assembly is screwed all the way down into the body. If you keep turning with the wrench, you can break the body or the stem. I know this because I ruined my mother's kitchen faucet in this way many years ago.
Photo of dripping sink by Son of Groucho/Flickr Creative Commons.