It depends on the type of faucet. Here's an article that will help you identify your type of faucet and walk you through fixing it. Wishing you the best.
Here's a site that tells about them:
Here's how to fix a leaky compression faucet:
A double handled faucet:
A single handle faucet:
Just a faucet:)
You should find something in there that will help you.
Betsy's answer pretty well covers it. First you need to establish where the leak is coming from (handle/handles or faucet). Then you need to know the make of your faucet (Moen, Standard ETC.) Most home improvement stores carry repair parts for faucets or you can go on line and get repair parts. And those repair kits will provide you with instructions which with the videos should help you out.