What kind of faucet do you have?
There are two main types of faucets - washer-type and washer-less:
Works with a rubber or composition washer that closes onto a metal washer seat. A faucet leak occurs when the washer becomes hardened and worn or the washer seat wears.
Washer-less, single-lever faucets
Comes in four basic types (disc, valve, cartridge and ball-and-cam) and are usually best repaired with a repair kit designed for the particular model that's leaking.
Please note: Internal components of your faucet are not the only thing that could be causing the leak. Calcium or mineral build-up can be a contributing factor to leaking, so removing build-up when repairing the faucet is a good idea.