Initially remove any obstacle from in the plug hole i.e. food, hair and other blockages. This can be achieved by using a suitable chemical cleaner or caustic soda. (Ensure chemical is suitable for the type of sink and plug hole you have e.g. enamel sink or chrome plug hole).
If this does not clear the sink then a sink plunger will be required.
Before ‘plunging’ cover the overflow with a damp cloth and hold it there, this will force the air to hopefully clear the blockage.
Place the plunger cup over the plug hole, and plunge vigorously up and down for a minute or two.
Should the plunging not clear the blockage then the next step is to remove the trap under the sink. A bucket should be placed under the trap to catch the water, which will come from the sink.
Check the trap for any blockage, if it is clear then a pipe cleaner (such as a flexible curtain wire) should be used to dislodge the blockage if possible. (See Clearing a blocked waste pipe)