There are many possible reasons why a faucet may leak. However, if water seems to be leaking out around the handle of your single-handle faucet, the culprit is most likely your o-ring. This is a rubber ring that sits inside the faucet handle where it connects to the base of the faucet. It's meant to create a watertight barrier as you operate the faucet handle, but it can become worn out and loose over time. Luckily, replacing the faucet's o-ring is a pretty simple process that you should be able to tackle yourself as long as you have some basic tools and patience.
Step 1: Turn off the water.
Find the cold and hot water valves that lead to the sink, and turn them both completely off so that you don't end up getting splashed with water as you work.
Step 2: Remove the faucet handle.
Most faucet handles are held in place with a single screw--usually a Philip's screwdriver. You may need to pop a metal cap off of the top of the faucet to reveal this screw. Once you find the screw, unscrew it and set it aside. Then, pull straight upwards on the handle; it should come right off the faucet assemble, revealing the valve stem.
Step 3: Remove the old o-ring.
The o-ring is the often black rubber ring that you'll see around the base of the valve stem. Use your fingers to remove this ring. You may need to slip the end of the screwdriver under it to loosen it a bit. Depending on your faucet's setup, it may have two or three o-rings. It's easiest to replace them all at the same time, so remove them all now.
Step 4: Put new o-rings in place.
You can purchase o-rings at most hardware stores. Take the old o-rings with you to the hardware store to ensure you purchase the correct size. Once you have the rings, slide them into place where the old rings were.
Step 5: Put everything back together. Slide the faucet handle back onto the valve stem, complete with its new o-rings, and then put the screw back into place. Tighten it, and then if needed, put the metal cap back onto the top of the faucet.
You can now turn your hot and cold water valves back to the open position, and test out your newly repaired faucet. With any luck, it should now be leak-free. If not, you may need to contact a plumbing contractor for more help.