The plunger, referred to as the "plumber's friend", will often clear up the clogged or slow drain. If you do not already own a plunger you can purchase one at your local home improvement store.
You will need to get the most suction as possible.
If you are working on a clogged double sink plug the other sink with a stopper or wet rag. (* plumbing tip put a thin coating of petroleum jelly on the base of the plunger to get a tighter fit and better suction)
Place the plunger over the drain and push down - pull back just a bit without breaking the seal. Do this 10-15 times then pull up quickly. If the water rushes down the drain you're done. If not try the process again.
You can repeat this process several times however, if the clog remains after the first few times it may be time to try something else. If you can locate your trap under the sink, simply remove the clean-out plug and pull the materials causing the problem from the trap.
This will require some tools and some minor working knowledge of plumbing. If you do not have access or do not want to try this on your own call a plumbing professional to assist you or simply try Carlsbad Plumber, AC Plumbing Construction