If the clog is located in a drain or pipe, you should first try to clear it using a standard plunger. If the plunger doesn't open up the drain, the next step is to get directly at the clog by using an auger or snake.
First and foremost, NEVER use a plunger if chemical clearers were added to the drainpipes. Splashing water can cause skin burns and severe eye injuries. Second, make sure that overflow drains are stopped up with wet rags BEFORE plunging. For sinks with two drains - cover up one of the openings and plunge the other. This ensures that the pressure applied with the plunger will push out the clog and not be released through the drain in the other sink. And third, once the overflow drain or other sink is plugged, run enough water into the sink to cover the cup, providing a better seal around the cup.
If plunging a couple of times doesn't clear the clog, you will need to open up the trap and get at the clog directly. Placing a bucket underneath the trap catches any water or debris when the trap is removed. Once it's open, an auger or snake can be slipped into the pipe to clear clogs further down in the pipe system. As the snake encounters resistance, slide the handle up and lock it, then spin it clockwise. Continue feeding more of the snake in and keep spinning until the snake moves freely in the pipe. When it does, simply pull the snake out of the pipe and into the bucket. There should be debris at the end of the snake. Once the trap and drain is reassembled, use the plunger on the sink again to force out any remaining pieces of the clog still in the pipes.