If you look at your bathroom sink near the top you will see a hole. This is the overflow and it prevents the water from rising up and spilling onto the top of your counter. When the water level gets high enough in the sink it goes into overflow hole and down to the drain. Nicer sinks have an integral overflow trough built into them that leads to the drain. These last and generally don’t leak. However, the inexpensive builder-grade sinks are different. Many of these sinks have a plastic trough stuck to the bottom of them that leads to the drain. This style does tend to leak over time.
You generally don’t have an issue with the cheap sinks though, since the overflow is not needed very often. You will only notice a leak when the water enters the overflow. The plastic trough really isn’t made to be repaired, but you can try to make it more water tight. Try caulking the perimeter of the trough, then take your finger and push the caulking into the area where the trough meets the underside of the sink. Then cross your fingers. With any luck this will hold the next time you leave the water running and the sink stopper in. The other remedy is replacing the sink, but for as often as the overflow is used, most people try to just “band-aid” the sink.