There are about four types of faucets in use today, they are the washer style (compression), ball, cartridge and ceramic disk. Your faucet should be one of these types.
All of these are pretty simple fixes so you're first job is to determine what kind of faucet you have dripping then you can begin the repair job.
Most washer styles (compression) are the two handle kinds like in your bathrooms.
The ball styles are the newer ones in your bathroom or kitchen that have only one handle that you push right or left and rotate or lift and push back to turn on. Most newer kitchen faucets are ball style.
The cartridge style is probably the kitchen sink with one handle that pushes right or left but lifts up instead of rotating on a ball.
The ceramic disk style is very similar to the cartridge type except that the insides are different.
Okay now that you have figured out what type you have let's go.
If you know the model of your faucet, you can call the hardware store and ask them if they have a repair/replacement kit on hand. If you don't and most of us don't know, then you'll have to take the faucet apart and bring in the insides for them to find one for you.
So to fix a leaky faucet, first turn OFF the water supply to the faucet.
Look under the sink for two valves, one cold and one hot. Even if you have separate hot and cold handles turn them both off just to be sure, then plug the sink drain. Many a tiny part or screw has fallen down the drain and thus precipitates a removing of the trap to get it back. Why create more work when you can prevent it by plugging the drain. No plug? Lay a towel in the sink to cover the hole.
There should be a "set screw" somewhere on the handle. It's usually located in the back where it's really hard to see and get to but it's probably there. It may be under a decorative cap on the top of the handle where the C(cold) or H(hot) is. Remove it and take off the handle.
COMPRESSION FAUCET/ WASHER STYLE
A new cartridge kit will be needed for this leaky faucet repair.
Remove the cap and unscrew the set screw. Take off the handle, and handle assembly and the spout. If there is a screw holding the spout, remove it too.
If you have a sprayer on the faucet then you'll have to remove a collar holding it in. Now remove the retaining clip (u-shaped clip).
Now remove the cartridge and clean out any build up or gunk from the housing. Replace the o-rings of the faucet just to be sure. Put some valve grease on the new cartridge and insert it into the housing.
Make sure that the notch in the new cartrdge is facing the sink bowl. Now replace the clip and reassemble the faucet in reverse order.
Check the faucet by opening the faucet about halfway and turn on the water supply. Let it run for a few minutes and if no leaks, you fixed a leaky faucet.
These types of faucets are very similar to the ball types. Remove the set screw and handle. Remove the ceramic disk assembly and clean out any gunk. Now replace the rubber seals
located in the spout assembly.
Reassemble the faucet in reverse order and test the faucet by opening the handle halfway and turning on the water supply.
If no leaks you just fixed a leaky faucet. If there are leaks then replace the ceramic disk.