I would suggest looking for an on-line video(like on YouTube) that does a pretty good job of showing as well as procedures typically required to fix it.
You should try to find the name of you faucet to help matching it to either a video and/or replacement parts.
Many are very commonly design but they do vary somewhat. I'll attach a link to a video that show rebuilding a Delta faucet.
They are not all like that. But if you can find a rebuilding kit made for your facet, that would be one way of doing it.
The other is to replace the cartridge, which is the entire plastic piece together with whatever sealing o-rings already assembled.
In my case I always buy faucets that come with a lifetime warranty. So whenever it starts dripping I just call them up and they send me replacement cartridges free of charge. I've been doing this for quite some time now.
Here's the link I mention for the Delta faucet. Gives you a pretty good idea of what to expect.
EDIT: Of course, if it's a one handle, (As
@jeffmattero76 asked) it would be a bit different but I sure you could find a video on it.