The first thing to do is shut off the water supply to the faucet that's leaking. There should be water shutoff valves located under the sink. In older homes there is often onyl one main water shut off valve for the entire house, usually located near the water meter where the water service comes into the house.
If there is only one main shutoff you will want to open all the other faucets to drain the pipes. If there is a shutoff close to the faucet you're working on you can skip this step. just make sure you shut off both the hot and cold supplies..
Close the drain or otherwise cover it to prevent any parts from being lost down the drain.
Remove the handle. Most handles are attached with a screw often hidden under a decorative cap. Use a screwdriver to pry the cap off.
Once the handle is removed use a wrench to loosen the retaining nut, remove the workings of the faucet and inspect the parts to determine what repair is needed. Some faucets have replacable rubber washers, others have cartridge assemblies that need to be replaced as a whole. Once you have the parts needed just replace everything in reverse order and turn the water supply back on.