Turn off the water to the kitchen sink; this usually can be done with valves on the supply lines under the sink. There will be a pipe down from the sink drain to a P-trap; the drain and strainer assembly will be attached to the sink above this. Remove the P-trap, which will be either copper with a chrome finish or plastic (polyvinyl chloride --- PVC) pipe. Metal fittings should be loosened with channel lock pliers (wrap the jaws to prevent damage to the finish); PVC can usually be loosened by hand.
Test the lock nut holding the drain assembly in place; it is just above the P-trap connection and around the strainer body outlet. If is loose, try to tighten it, using channel lock pliers while holding the drain itself in place from above. Some plumbers insert screwdrivers in an X fashion to hold the drain while tightening the lock. If that nut is rusted and will not tighten, you will have to remove it. Often it will break rather than unscrew.
Remove the drain and strainer assembly, then clean the surface where it fits the sink. If the drain is reusable, you can reinstall it. Otherwise, buy a new drain assembly (these should be standard fittings). Run a bead of plumber's putty or silicone sealant around the bottom of the drain area. Then fit the new drain in place, with a new gasket and washer between the locking nut and strainer body. Tighten the locking ring, using channel lock pliers and holding the top firmly in place (it may help to have an associate hold the drain). Then reconnect the P-trap, turn on the water supply and test the drain for leaks.