Somewhat depends on the design. With any luck, there should be a
couple of shut off valves (hot and cold) under the sink. Turn these
off. Typically, the faucet that moves is held with a retaining ring
that screws on. There might be a set screw or two as well.
Under this should be a couple of "O" rings. Get a small tube of
silicon grease (any good hardware or plumbing store). Clean things up
with a tooth brush, lube the "O" rings, and the threads of the
retaining ring. Put it back together. That's it. Watch out, things may
be a trifle corroded, which might be adding to your woes, and you
might not be able to get things apart. In that event, replace the
whole faucet assembly, especially if you are in a hard water area. If
you do replace it, grease all the threaded fittings. Makes it MUCH
easier to take apart in the future. Home Depot or Lowes is your friend
here. I have found that it is not worth the effort to try to service
these things, but simply easier to replace them.
Let us know what the outcome is. Especially what make. What I said
applies to Delta, IIRC.