Run water until it is several inches deep in the sink.
Locate your sink's overflow hole. This allows water to drain rather than overflow onto the floor.
Wet some toilet paper and squeeze it until solid. Then shove it into the hole, creating a plug and stopping any air from getting into the drain.
Get you plunger (or mini plunger).
If your plunger looks like this, push the ring inward.
It should now look like this.
Plunge hard and fast. Most importantly, do not quit. Too many people either don't put enough effort into plunging it or give up after 5 plunges.
Make sure that at least 2 inches of the plunger is submerged in water and that the sinks overflow hole is sealed. If air is still getting in, hold the tp plug in the overflow hole while plunging.
Plunge until the drain is unclogged and water drains quickly. Most of the time the bathroom sink is clogged with hair, using chemicals may work but Do Not use them if you have metal pipes.