Using a plunger

The secret to successfully unclogging a toilet or other drain with a plunger is suction.

Set the plunger over the drain, push down and make sure it's set all around the edge of the plunger. Then pull up vigorously to dislodge the clog. Repeat until the water drains. You may have to raise the water level of the sink or toilet to cover the plunger to see that it's sealed to the surface around the drain, and also to see when the clog is dislodged.

A bathroom or vanity sink frequently has one or more overflow ports up on the front of the sink to prevent the sink from flowing over the top. These must be sealed off or you won't be able to develop any suction. Soak a rag and stuff into the overflow hole(s) to seal them.

Sometimes the clog is just fibrous material mixed with some black gunk that is probably mold that grows to hold the hair or other fiber together. An easy solution to this problem is to pour a gallon of bleach down the drain and wait for it to kill the mold so the fiber can wash down. Just get a gallon bottle of the cheapest bleach you can buy (usually not more that about $1) and pour the whole thing in. It should take only a few minutes, 10 or 20 at the most.