When an elbow pipe under the sink is old and dripping it can be a real nuisance, as well as damage whatever is beneath it. A clogged elbow can cause problems as well. The lower part of the elbow pipe, sometimes called a P-trap, is a catch-all for sink debris and once it gets full, drains run slow and stopped up drains are more likely to leak. No need to call a plumber though, an elbow drain can be changed by anyone with basic tools.
Bail out the sink into a working bathtub or into a bucket. You do not have to turn off the water supply for this job, just make sure no one turns on the faucet of the sink while you're working. Be sure you have the right diameter new elbow. You may decide to remove the old one first and then take it to the hardware store to be sure you match it up correctly.
Loosen the upper and lower nuts connecting the elbow to the other pipes. Try your large pliers first, especially if the pipe is PVC, but if they are not large enough or strong enough, a pipe wrench will usually loosen them quickly. Metal elbow pipes usually require the pipe wrench, although you may have success with a strong pair of larger pliers.
Put your rubber gloves on just to be sure you don't pick up any bacteria on your hands from the liquid coming out of the old drain once it is disconnected. Place an old dish pan under the pipe. Remove the old elbow from the pipes and dry the threads on the pipes on either side of the old elbow with an old rag.
Wrap the threads of the pipes the new elbow will attach to with the plumber's tape to reduce chances of leaking. Put the new elbow in place and hand tighten the nuts on both ends. Be sure that the threads start out evenly and not on an angle. The nuts should hand tighten easily. If not, unscrew and realign with the threads until it screws on smoothly. Once hand tightened, tighten the rest of the way with your pliers or pipe wrench.
Test the elbow for leaks by running water into the sink above while you are watching the elbow. You can use tissue paper also by dabbing it around your new connections or placing it around them so see if any moisture is picked up. Fill the sink up half way with water and then let it drain out all at once. This force will find a leak if you have not tightened the nuts tight enough or if they were uneven on the threads. If the nuts are just not tight enough, retighten. If they didn't go on straight to start with, loosen them and start over, being sure they are going on straight.