One of the simplest ways to clear a clogged bathroom drain is to use a plunger. Just place the rubber cap of the plunger over the drain opening, ensuring that the opening is entirely covered, and dispense a number of sharp downward thrusts.
Creating a series of compressed air blasts in this way often dislodges the offending blockage. If you are working on your sink, make sure you block the overflow outlet first. If left open the outlet will allow the compressed air to escape rather than exert pressure on whatever is clogging the pipe.