Step 1: Check the brand of faucet you have and determine if you can get parts. Go to the nearest hardware store and check to see if they carry parts for your brand of faucet.
Step 2: If you have kept your instructions on the faucet either when you purchased your home or when you or someone installed it years ago your steps ahead of the game. Why ? Because it makes it easier to view a picture of the internal parts and how to get the faucet apart without breaking parts. Also many of the newer faucets have a ten year warranty and you may get a new faucet back if you kept your paperwork.
Step 3: Get your tools ready. You will need a handle removal tool, 10 or 12 inch crescent wrench, set of Allen wrenches, channel locks, Phillips screwdriver, regular screwdriver, Wd-40 or some other similar rust and corrosion removal spray, stem lubricant, socket set, and depending on your faucet you could need many other tools.
Step 4:Turn your water off and begin by taking the handle off your faucet. This could be a shower faucet, lavatory faucet, kitchen faucet, or any type. You may need to spray the bolt with WD-40 first to help loosen the bolt or Allen screw before you attempt to remove the handle. If you strip the bolt you will need to get the bolt out one way or another before you can proceed.
Step 5: Take off the lock nut that holds the stem.
Step 6: Remove the stem or cartridge. You may have to spray all around the stem to loosen it. With shower valves you will need to have a socket set to remove the stem. Feel the back of the faucet with your finger and sometimes you can determine if you have a bad seat. If the seat is not smooth to the touch you will have to replace it. I recommend that you replace the seats anyway since you have the faucet apart.
Step 7: If you have a seat you will need to remove it with a seat removal tool.
Step 8: Replace the seat and either rebuild the stem with the proper o-rings and other parts or a whole new stem. Insure you lubricate the stem with stem lube before you put your new stem back in place.
Step 9: Put all the parts you took off back together the same way you took them off.
Step 10: Turn your water back on and check to see if you no longer have a drip. If you do I suggest you repeat the process at least one more time to double check yourself that everything was done correctly.
Step 11: If this all fails you may have to replace the faucet.