1) Turn off the water. Before you start, turn the water off to your faucet. Look underneath your sink for the pipes that run up to your faucet. Along those pipes somewhere will be handles that you can turn to shut off the water to your sink.
2) Turn the handles clockwise to turn off the water. Turning the handle right tightens the handle, and thus turns your water off.
3) Plug up the sink drain once the water has been turned off. Use either the sink's in-built mechanism for plugging the drain, or plug it yourself with a rag. Nothing will ruin your day faster then having a screw or a washer going down the drain.
4)Take the faucet handles off. Take out the screw that is holding them on. Use the screwdriver for this. Some faucet handles might hide their screws behind plastic or metal caps. You might have to pry the cap off with a flathead screwdriver to get at the screws holding the handles on, or unscrew a cap. Once you've removed the screws, pry the handles off with a flathead screwdriver. Some handles might come off easily without the need for prying.
5) Once you have the faucet handle off... remove the old washer that was behind the handle. If there is no washer behind the handle, that is okay, you are going to add one.
6) Take the washer you've purchased. Simply place the washer where the old washer was. Basically the hole in the washer should be just slightly bigger than the post that the handle turns.
7) Put the faucet handle back on the post and put the screw back in. Tighten the handle until it is "hand tight". That means tighten the screw until it feels tight without you having to use extra strength to tighten it.
8) Turn your water back on by turning the handles under the sink counter-clockwise
9) Check it. If everything went well you shouldn't have a leaky faucet any longer. If your faucet continues to leak, try tightening the screws on the faucet handles just a bit more. If the faucet still leaks, then it really is time to call a plumber.