Leaks in plumbing are common, especially in older homes. Water dripping or pooling under the sink doesn't necessarily mean you need a plumber. Many home plumbing problems are easy to learn how to fix yourself. The most difficult part of a home fix is finding the exact problem. A leak under the sink could be poor caulk around the faucet or sink itself, a damaged p-trap, cracked PVC pipe, leak in the wall and a host of other issues. The problem can be narrowed down pretty easily by drying off the area and using a flashlight on different locations. The most common problems are the p-trap or a bad pipe connection.
Turn the water off at the valve under the sink. Position a flashlight under the sink so you can see what you are doing. Place a bucket under the p-trap and loosen the nuts with an adjustable wrench. The p-trap will probably have water in it so make certain the bucket is positioned correctly.
Wrap the threads on the end of the p-trap and the drain pipe with plumbers tape. Wind it around two or three times. This will ensure a tight seal when you tighten the slip nuts. The plumbers tape is not really sticky so it will be easy to remove if you need to replace a pipe.
Tighten the slip nuts completely. Turn the water back on. Turn on the taps and check under the sink for leaks. If you see anything it is likely the nuts are not tight enough. Re-tighten and check again.
Seal the sink and the faucet plate. To seal the sink, run a small bead of silicone around the periphery of the sink. Use a smooth even motion. Then, run a rag around the sink to remove any excess. The face plate can be pried up if it is not sealed. Dot on plumber's putty around the edges of the plate and then re-affix. Wipe off excess putty.
Turn off the water and place a bucket under the plumbing. Remove the p-trap and the tailpipe that comes down from the drain. To remove this, put a basin wrench on the nut under the sink and a pair of pliers through the drain on the top. This is a two-person operation. Each person twists in a different direction to loosen the nut.
Disconnect the pipe that attaches to the wall. Pay attention to what pipe goes where so you can reassemble. Check every pipe for cracks or breaks. If you find anything, you will need to replace the pipe, too.
At every thread site wind some plumber's tape. Reattach the pipes using new washers and nuts. Tighten everything completely. Turn on the water and check for leaks. If there are still problems, it may be a leak in the wall.