The sink that came with the house was an early, but somewhat plain pedestal sink that was badly rusted out.
Although I had intended to restore it, the truth is I can replace it with another antique sink much more cheaply - if it were original I'll shell out the cash, but I doubt it is. To fulfill our 203k obligations we had to make this bathroom operational, so we picked up another vintage pedestal sink with working taps on Freecycle to serve in the interim. Those taps had started to fail, and the bad refinishing job on the sink (not our doing) was only getting worse, so when I saw this sink for $50 on craigslist I jumped on it. Of course it wasn't the sink I was interested in, it was those lovely nickel taps...
Remove the handle by unscrewing the screw on the end of it (yours may have a cap that needs to be popped off), then use a wrapped wrench to remove the nut.
Take pictures of what you find, usually a decayed washer and/or packing if there have been previous repair attempts.
Once we got the stem out we cleaned everything, and tracked down the appropriate washers/screws. Skip the big box stores and go directly to a plumbing supply if you have a truly old faucet like we did.
Neatly wrap the packing
If there are any other leaking connections unscrew and pack/use thread dope as necessary.
Smile, you're done. Take an "after" picture that doesn't show the item in question...