If you know what type of faucet you have, you are ahead of the game. But if you do not, you will need to take your faucet apart first to see what is going on in there. Especially in older homes, the most common type of faucet is a compression faucet. This means that inside the faucet there is a rubber washer that helps to create the seal when the faucet is tightened.
When faucets like this leak, chances are it is simply because the washer has worn out and needs to be replaced. Before you can start working, you will first need to turn off the flow of water to your sink. There should be a couple of knobs under the sink that allow you to do this. And be sure to cover or plug the drain before you begin so that you do not accidentally drop anything down it as you are working.
Once you have set yourself up, unscrew the faucet handles (or whichever one is leaking) and remove the outer part. At this point, you should be able to see the washer. Simply remove it and replace it with a new one and then put the faucet handle back on and you should be as good as new. These types of washers tend to wear out from time to time, so you will probably have to do this again in a few years or so.
When you are replacing your washers, though, it is important to remember a couple of things. First, if you are unsure about what type of washer to put in your faucet, take the old one to the store with you and ask a salesperson for help. You want to make sure that the washer you put in is exactly the right fit for your fixture so it is worth taking the extra step to ensure that you have the right piece.
And after you have taken apart the faucet, it is a good idea to check the stem to see if that has worn out and needs to be replaced as well.