Shut off the water to the faucet. Look under the sink, and find the shutoff valves. There should be one valve for each the hot and cold supply lines that attach to the underneath fixtures on the hot and cold handles. If it is an older house, the shutoffs may be under the floor line and may need to be accessed from the basement.
After making sure the water is off, remove the caps on top of the handles by gently prying them off with a flathead screwdriver. Remove the screw that holds the handle in place, and pull off the handles.
Remove the stem by loosening the retaining nut that holds it in place using the adjustable wrench. Gently remove the stem with a pair of pliers. This process may take some time and a little gentle persuading, but it should eventually come out.
Remove the seat and the spring that will be left in the cavity once the stem is removed. The best way to do this is to use a screwdriver and gently pick out the seat and spring with the tip of the screwdriver. Before installing the new parts, smear them with teflon grease and put some teflon grease in the hole. This will help the parts work better for a longer time. Place the new seat and spring on the end of the screwdriver, and place them into the hole.
Replace the stem and hand tighten the retaining nut. Use the adjustable wrench to snug the retaining nut. Be careful not to over tighten it. Replace the handle by tightening the Phillips head screw and pressing the cap back on the top of the handle. Turn the water shutoffs back on, and enjoy a faucet that no longer leaks.