If the drain is clogged then you can plunge it or buy drain openers. If the sink leaks around the drain then the rubber ring needs to be replaced. If you can't get the sink unclogged then remove the elbow of the drain and wash out.
If the drain itself is leaking most likely you need to replace the rubber between the sink and drain. It also can be leaking from the trap or even from the shut off valves.
