The most common culprit for a leaky faucet is a bad washer.
Tools needed:
A screwdriver or two - maybe one flathead and one Phillips
You may need some Hex Keys
New washers (maybe take the old one off ahead of time and take it to the store to be sure you're getting the right size)
Preparation (It almost goes without saying):
Turn the water off first
Look under your sink and see where the pipes run. Somewhere along the water lines, you will see either a handle or you can shut off the main water line, usually located in your basement near your water heater.
Block the sink drain to prevent an accident like a screw or a small nut dropping down the drain.
It goes without saying - make sure you have bought a replacement washer (or two) from the local hardware store.
Time to get to work
Locate and remove the screw that is holding the handle of the faucet in place (it may be hidden behind a small plate or cover and that cover may need to be pried off).
Time to remove the handles. Use a screwdriver to do this - you might need to use a little bit of force to pried it but it will come.
Now you will see the old washer. Take off the old washer and replace with the new. Notice that the hole in the washer should just be a touch bigger than the central post that turns the handle.
Now re-tighten the handle, but only hand tight until it stops turning without extra force needed to be applied.
Next, turn on the water again and see what happens! If there's no leak you're good to go! If it still leaks, try tightening the handle a tad bit more.
If it still leaks after that- then the problem is elsewhere, but at least it was worth a try!