There are two types of plumbing under the sink. The first type brings water into the faucet when you want it. The second takes away the water so the sink drains. You can replace most of the plumbing under the sink if you have specific tools. If the waste pipes need repair, you simply take the old part and put in a new one. The problem with all plumbing is that if you don't do it properly, you'll end up with a leak.
Find out whether you need to repair the incoming water plumbing or the outgoing water plumbing. If you notice a leak, you might not be able to locate the source, primarily because water travels. Dry off the pipes under the sink and plug the sink. Run the water until the sink is full. Check under the sink and see if you have a leak. If you do, most likely it's the incoming water tubes. If you don't, turn off the water, let the water out of the sink from before and watch under the sink as it goes down. You'll find the leak it it's there.
Shut off the water at the shutoffs, the little handles on the skinny pipes, if the leak comes from water going into the sink. There are several leak possibilities here. If the plastic tubes (the water supply hoses that go to the faucet) have a crack, it causes a leak. The shutoff could be bad. The connection to the faucet (the coupling nuts) might be loose. Locate the spot that seems to leak before you shut off the water. The highest damp point is usually the spot.
Tighten the nuts with a wrench if they feel loose. Sometimes this is all you need to do.
Loosen the nuts at both ends of the water supply hose if it leaks. You'll need a basin wrench to loosen the nut at the top. It has a floppy head that allows you to get the wrench around the nut in a tight spot. Take the hose to the home improvement store with you to buy the right-size replacement. Insert one end in the water supply fitting and tighten it down. Put the other end into the faucet and tighten the nuts.
Take off the old shutoffs if they leak, but turn off the main water supply first. Sometimes these just wear out, although it doesn't happen very often.
Loosen the nut at the top of the shutoff that connects the water supply and then loosen the nut at the bottom that fits it to the supply pipe. Hold the pipe still with a pipe wrench and use a smoothed jaw adjustable wrench to turn the nut. Take it to the home improvement store to get a replacement. Reassemble by first putting on the shutoff and tightening to the pipe, then inserting the feed tube back into the shutoff and tightening.
Look for a loose connection in the waste pipes. Tighten all the connections using two groove joint pliers--one to hold the pipe and one to tighten the nut. Don't overtighten it. It will break.
Take apart the plumbing if that doesn't work. Sometimes you just need to replace a washer to fix the drainpipes. Get replacements for worn parts like the plastic ring washer or compression nuts. These are inexpensive. If you see a crack in waste pipe take it out and get a new piece to match. If you need to cut it, use a plastic pipe saw. Replace the parts as they came off. Don't overtighten the nuts.
Use Teflon tape on all threads before you put nuts back on to the threads. It makes removal easier and provides a little extra barrier to resist a leak.
Use only plastic washers on plastic pipe and rubber washers on metal pipe, or you won't get a seal.