There could be many reasons why a faucet will not turn off. You will need to identify the faucet in order to be able to get replacement parts. Sometimes this is easy with the brand name marked on the faucet and other times you may have to bring parts with you when you go to get replacement parts so the clerk at the hardware store or, plumbing supply house can identify the part. If you know what the brand is many times you can get parts diagrams on faucet company websites that show the parts and numbers which also will give you an idea of how the parts come apart. Usually faucets are relatively simple and a person with some mechanical skills and a few basic tools may be able to take a faucet apart to find the bad part that needs replacing.
The faucet not shutting off may be as simple as a handle that is stripped out so the handle is spinning but not moving the stem underneath. Handles are joined to stems with fine-toothed splines making the connection. These may corrode and strip out which in most cases can be fixed by replacing the handle. Sometimes you may need a new stem as well. Sometimes you will need a faucet handle puller to remove the handle.
When fixing a faucet that does not shut off it is good to know which side you are dealing with so you can pay closer attention to the bad side but it is generally a good idea to renew both sides at once.
Before going any further after removing the handle you need to shut off the water to the faucet. Usually closing the angle stop valves under the sink can do this. You should open the faucet afterwards to check and make sure the water is turned off, and leave the faucet open when taking it apart further.
The next thing you will need to do is check the condition of the stems or cartridges to see what is wrong with your faucet.
Two handle faucets may have stems and seats, which screw out of the faucet body for repairs and replacement. Stems should be checked for wear and replaced if needed, or, they may be lubricated and have the washer replaced. The seat requires a seat wrench to be taken out and should be replaced or, at least sanded smooth. There will also in many cases be a second nut on the top, which tightens the packing. When you put the parts back together the packing will need to be adjusted just tight enough so it doesn’t leak.
Some newer two handled faucets will have cartridges or, stem packs which are not repairable and should be replaced. Pay attention to the markings on the cartridge or, stem packs to make sure the replacements get installed in the same orientation or the handles may open and close in reverse.
Single handle faucets are usually held together with a single nut, screws, or, a clip on top. Some like Moen use a cartridge, which will need to be replaced requiring the use of a Moen cartridge removal tool. Others like a Delta may use a ball and rubber seats and springs. The Delta ball fits into a pin, which you will need to be very careful that you do not break or, you will need a new faucet.
In many cases if a faucet is very old and in poor condition it is better to replace it than waste a lot of time on it. If you are not sure of your abilities or, not wanting to invest in specialty tools you may just want to consider hiring a plumber.