Slow-moving or stopped-up drains are common in bathroom sinks, but luckily the fix is usually simple and takes only about 15 minutes. The problem is caused by hair and gummy soap scum that get caught on the stopper or pivot rod and clogs the drain.
To get at the clog, try lifting out the stopper (Photo 1). Sometimes it'll come right out. If it doesn't lift out, it's held in by the pivot rod. Release the stopper by removing the pivot rod nut and pulling out the pivot rod (Photo 3). If you can't loosen the nut by hand, use pliers. With the pivot rod pulled out, you'll be able to lift out the stopper. Then to get the clog out, bend a wire in a tight hook (a light-duty clothes hanger or short length of electrical wire will do) and fish out the hair (Photo 2). If you didn't have to remove the pivot rod to remove the stopper, you can just drop the stopper back down into the drain. If you removed the pivot rod, first drop the stopper into the drain. Then line up the pivot rod with the slot in the stopper and reinsert it. Finally, hand-tighten the pivot rod nut.
Run hot water down the drain to help clear out any remaining soap scum and to check that the clog is gone. Check around the pivot rod nut to make sure it's not leaking. If you see drips, tighten the pivot rod nut slightly with a pliers.