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These faucets are usually very easy to repair. If there is a brand name on the faucet, make note of it for your trip to the hardware store later. I will try to give a general description of the repair, though your faucet may vary in part. The repair sequence is as follows:
1) Turn off the water under the sink. Then turn on the faucet on (hot and
cold sides) to be sure they water is indeed off and also to relieve pressure in
the lines so you don't get "sprayed" when you disassemble the faucet.
2) There is usually some sort of cap on top of the faucet handle that conceals a screw. Pry this cap off with a screwdriver or other flat instrument. Sometimes a pointy instrument is needed, such as an ice pick or awl. Loosen and remove the screw under the cap and pull the handle off vertically.
3) Under the handle you will find a nut that holds the faucet stem or cartridge in place. Unscrewing this nut will release the stem and allow you to pull it out. It may not come out easily; it can be pried out or pulled out with a pair of pliers on the stem. This can be the most risky step, since if the stem breaks you might have a hard time removing the reset of the body.
4) This step is brand specific. You will either find a washer underneath the stem in a depressed area of the housing, or maybe not. Delta style faucets, for example, have a cylindrical washer with a spring underneath. Be sure to note the orientation of the spring… the smaller end is installed up, not down. If there is no washer, you may have to replace the entire stem. Perhaps the best advice I can give you is to do what I always do when dealing with an unfamiliar fixture… take the parts to the hardware store or plumbing supply house to get exact matches.
5) Reassembly is in the exact same order... but opposite! Good luck!