Plungers not only work well in toilet clogs, but also clogged drain pipes. You may want to consider investing in a separate plunger for your drains for sanitary purposes. Small plungers that are designed for use on sinks and drain openings are widely available at home improvement stores. Place the plunger over the opening of the drain, and push and pull vigorously for about 10 seconds. Rinse hot water down the drain and repeat as needed until the drain empties properly.
Browse the cleaning product aisle of your supermarket or hardware store, and look specifically for the canned products that use air pressure. Typically, you are instructed to plug the drain and fill the sink or tub with a few inches of water. Then you pull the plug and place the can opening over the drain as the water drains. Press the can down firmly and hold for a few seconds, then let the remainder of the water drain out of the sink or tub. Turn on the water supply and examine drain flow. Repeat the air pressure application until the drain empties quickly.
Pumps are a bit different than the miniature cans found at the supermarket, and these are typically more readily available at hardware or home improvement stores. These come in a variety of sizes and price options, and all work the same way for the most part. The open end of the pump is placed over your drain and then a trigger is pulled at the opposite end for one minute or so. This vigorously pumps air pressure down the drain.
When you are unable to clear the drain on your own, consider contacting a plumber for professional assistance. Plumbers can use augers that wind down the drain and pipe snakes that are pushed through the drain. They even have the capability to professionally examine the pipes, take apart any portions where a clog may be present, and clear the drain accordingly.