A single handle faucet that leaks around the base may be an easy fix or you may be replacing the faucet. If it was a kitchen faucet, I would replace the o-rings, but the difference is a bathroom faucet is that the spout doesn't swivel, so those o-rings don't exist. There are a few things to look at though.
See if you can tell where the water is coming from. In many cases, the water will be traveling from the aerator, down the underside of the spout and finally puddling around the base. If this is the case, remove the aerator and either clean it up and reinstall it, or just replace it.
Another possibility is that water is coming from around the handle. In this case I would replace the cartridge. You will need to turn off the water, remove the handle, remove the retaining pin and finally pull out the cartridge. Get a replacement and reinstall it.
Lastly, it may be that the faucet is corroded and needs to be replaced. You may not even see the leak from above. The body of the faucet may be corroded and pitted or cracked and you may not see it. If you can't see the leak, you probably are looking at replacing the faucet.
It always seems to come down to the price, whether you are buying a faucet or a widget. So if you are buying a faucet, should you buy a quality popular brand (Moen, Delta, Price Pfister, etc.) or a cheaper imitation?
It is usually the case that you get what you pay for, and in my opinion that certainly holds true for faucets. I once installed a brand new faucet that was a cheap imitation. I paid 60% less for that faucet than if it had been a name brand. It leaked as soon as I turned the water back on…a seam in the faucet body was leaking and unrepairable. I blew an extra hour of my day replacing it with a quality faucet.
As a person that performs repairs on faucets, it is all about the parts. Cheap faucets have tons of cheap plastic parts. Although the styling is similar, you can feel it and it is inferior with plastic where chrome should be. Name brand faucets usually offer lifetime guarantees and if you contact them they will send you replacement parts for their faucets. For free.
When a problem develops with a cheap faucet, the parts are more difficult to get. In some cases you need to go to a plumbing supply company and possibly order the parts. With the hassle and time involved, you start to consider if it is even worth it. Since you paid so little for the faucet, maybe you just replace it (again). Only this time, you might consider replacing it with a major brand.
When choosing a new bathroom faucet, you can choose the deck mounted faucet or the widespread faucet. The difference is on a widespread faucet you can see the countertop between each handle and the spout. On a deck mounted faucet, the handles and spout are all attached to the body of the faucet. Widespread faucets are cleaner looking, but also more expensive. They are also more challenging to install.
Each handle and spout will mount independently of each other. As you place each through the holes in the sink or countertop, use plumbers putty under each and secure each with a series of washers and nuts. There will be a handle specifically for the cold side and the hot side.
After securing the spout, you will install a tee fitting on it so that the holes in the fitting are oriented to the hot and cold handles. There will be a series of gaskets, o-rings and nuts that hold this fitting underneath the spout. You will connect each handle to the tee fitting under the spout with hoses. Tighten the hoses but don’t allow them to kink. You can loop them to avoid kinking them.
Finally connect the water supply that feeds each handle valve. You can use a standard water supply line for this from the angle stop to the handle valve. Connect the sink stopper to the faucet pull rod. Tighten all connections and turn the water back on to test for leaks.
The other day we were working in a 50 year old house whose bathroom faucet at the bathtub would not stop running. This is a case of replacing (or rebuilding) the stem and seat. The stem is attached to the handle and rises up and down. At the end of the stem is a washer that pushes against a seat to stop the flow of water. Over time, this washer will wear down, or the seat will develop pits that won't allow it to stop the flow of water anymore. These parts need to be repaired or replaced.
So after you turn the water off to the house, prepare for some fun.
1. Remove The Old Handle
This isn't as easy as it sounds. Usually a screw is removed and the handle pops right off. Well in older plumbing, the handle can seem like its welded to the stem. In this case a handle puller can help remove the handle...or break it trying.
2. Remove The Old Stem
Removing the old stem can be a leap of faith. A deep well socket and a breaker bar will usually extract it, but you just hope that nothing else breaks. Once we got the old one out, it was time to go hunting. This stem was nasty and difficult to identify. After searching, we were not able to find a replacement for it. Instead we bought new parts and rebuilt it. But that only solves half of the problem.
3. Replace Or Resurface The Seat
The faucet seat screws into the faucet body and has a hole in the center of it. In this hole you insert a seat wrench to remove the seat and replace it with a new one. If you can't remove the seat, you can use a seat resurfacing tool to smooth out the surface so that the stem washer can stop the water.
Once you reassemble the bathroom faucet, the it should operate as it did 50 years ago. Many modern showers use a single handle to control the hot and cold. Here is how to replace a shower cartridge.