Many vanity and tub/shower faucets have rubber parts (washers, o-rings) inside the main valves that wear out every few years. If you find that a faucet still drips when turned off, then you likely have some rubber washers and/or o-rings that need replacing. The first thing to do is turn off the water to these valves. For tub/shower applications, a lot of newer faucets have integral water shut-offs. To determine if yours does, unscrew the faucet handle and remove the metal plate covering the vale body in the wall. You may find two valves without handles that can be turn clockwise with a standard screwdriver to the off position. If that is the case and you turn those valves off, you can then safely work on the main valve body without getting sprayed with water. If you do not have these shut-offs right before the fixture, you must find the water shut-off to the unit or home and turn off the water there. Also, you need to turn off the hot water valve coming out of the top of the water heater. Once you have done this, relieve pressure in the water lines by opening both a hot and a cold faucet valve. Once you have done this, you may unscrew the valve/s and take those parts to a plumbing store for replacement parts. Be careful to keep it all in tact as you may have to re-install it without fixing it that day if you can not locate the parts and need to order them. Remember to write down the information you have about the fixture to take with you to the store. Sometimes is makes sense to call the manufacturer and order parts by mail over the phone in advance as they are usually not expensive.
The benefits to fixing these common sources of water loss are obvious. You will save a lot of money and a natural resource will be conserved for when it is needed. If you are on a municipal water system, your bill could have gone up in the last billing cycle by hundreds of dollars (note that if this is the case, most water departments will refund that money if you show a copy of your materials receipts with an explanation of what you did).
WHAT DO YOU NEED TO DO IT? Most of these corrections require a few hand tools like pliers and a wrench, a standard flat screwdriver, a cross-head (Phillips) screwdriver, a caulking gun with some clear or white silicone sealant, and some patience. It’s a also a good idea to be aware that you made need a hardware store to be open the entire time you are doing the work so start in the morning just in case things do not go as fast or smoothly as you hope.