Using A Snake
A plumbing snake is a long piece of flexible metal that can be manoeuvred through your pipe to clear a blockage. It is sometimes called a cable or pipe auger. Before you start, put a bucket under your sink, shut off your sink's water supply, and take out the trap. The trap is that U-shaped pipe under your sink.
Push the snake carefully into the open drainpipe. Move it back and forth to help it get around all of those turns. When it reaches the blockage, you should feel it. Keep twisting the snake until you feel the pipe completely clear. If the blockage is a mass of hair or something similar, try to pull it out, because it probably won't break up effectively.
If you can't remove the trap, you can feed the snake in via the sink. Just keep guiding it around until it hits that blockage.
If you can't get the snake in to the pipe through the sink, push it through the trap cleanout. The snake should be directed towards the sink, or towards the wall, depending upon where the clog is.
Tip: If your snake just isn't doing the job, you may have a bigger problem than you thought. Your Main Drain cleanout or Soil Stack could be the source of that problem. In order to find out if this is the real culprit, take a look at the other drains in your house. If you have a few drains backing up on you, than you probably have a problem somewhere deeper in your plumbing system. In this case, call in a professional.