The most common cause for a faucet to leak is a worn out washer. Doing the repair yourself will save you on plumbing costs. For this task, you need to keep some spare cloth, screwdrivers, towels, newspapers, and a knife on standby. Cover the floor using towels and paper, so that the water does not collect on your flooring.
Blocking the shower drain to avoid small parts like screws, nuts, rings, etc., from flowing away. Cover the drain and flooring with a towel or paper. Normally a faucet will have a plastic or metal faceplate which will need to be removed. This can be popped off with the help of a knife or a screwdriver.
After you remove the faceplate, you will see a small screw. Remove it gently, and keep it aside, as you will need it later. Dismantle the escutcheon, which is the face plate around the fixture, by simply popping it off the wall.
Get a pair of pliers and locate a nut or pin securing the cartridge in place. Using the pliers remove this by turning the nut or popping off the pin and keep the smaller parts aside. Pull the cartridge out of the wall very carefully to prevent any damage.
Check if the cartridge is damaged or is attacked by mildew. Consider cleaning the cartridge under running water. If the damage is irreversible may be you will need to replace the entire unit. If it is not, then simply remove the O-ring made of rubber. Remove the old one and fix a new one in its place. This will most probably solve your leaking problems.
Finish off by fixing everything back in reverse order. Fix the last thing you dismantled for reassembling the whole faucet ensemble. Again cleaning up the faucet at this point will be a good idea to keep it free of dirt, mildew, and grime. Once you reassemble the unit, test it by turning on the cold and hot water to make sure that there are no leaks.
The method might slightly change depending on the type of the faucet, but overall, the method for any single handle faucet will be the same. For instance if it is a disc faucet, you would need to open or the unscrew the cap covering the cartridge. Lift the screws out and there will be a set of seals at the bottom. Just take them out and replace them with the new ones.