A dripping shower or sink faucet is a very common household problem. Here's my take on the best course of action.
I first check for anything obvious. A loose handle or nut. Sometimes a turn of the wrench is all it takes to stop that irksome drip.
Most times though, it's not that easy. There are several internal parts to a faucet, there are washers, seals, grommets, etc. Any one of which could be the source of the leak. Most manufacturers sell rebuild kits that include the most common internal parts that need replacing. This is easy enough, just involves a little disassembly and patience.
The hard part is figuring out what manufacturer and model faucet you have. They are rarely labeled and if they were at one point, that marking has likely been worn off. I've had good luck with google image search in these cases. Just take a picture of the faucet and then do a reverse google image search. You'll usually get lots of hits for matching images and will then be able to determine it's specifics.
Lastly, and sometimes the easiest solution is just to replace it. Typical bathroom faucets start at about $25 and replacing one is less involved than rebuilding one.
Either way Professional Handyman can help.