Sometimes, when a wash-basin or sink has clogged the best solution is to call a plumber who will help us solve the problem, but perhaps you might want to try some things first.
The cloggings in wash-basins or sinks normally occur as the result of a mass of organic matter stuck between the strainer body and the trap, without reaching the tailpiece. The first step is to remove the clogging from the wash-basin and to try to clean the strainer body with a curved wire.
If the clogging persists the problem can be in the trap. Perhaps the clog has got stuck in it. In that case it is convenient to remove the bottom part of the trap. Watch out; do not to forget to place a bucket under it as some water will come out. As soon as the trap is open use the same wire with a bent end to clean the inside.
If nevertheless the clogging remains or the trap is not detachable, you can resort to the use of a plunger: fill the wash-basin with water and place the plunger so that it covers the strainer body completely. A few energetic movements should displace the obstruction.
If even so you do not manage to unclog the wash-basin you will have to resort to the use of some unclogging product sold in drugstores and DIY shops. It is necessary to be cautious in the use of these products as they can be very harmful to your health. If the clogging is of organic origin, it will end by dissolving and with it your problem.
If despite everything the problem persists, perhaps it is in the tubes, so the best thing is to call a good professional: a plumber.