Cleaning the u-bend/trap
If you can't clear the blocked sink with the plunger the next step is to take the u-bend or trap apart. Modern plastic traps are easy to take apart without tools as they unscrew.
Put a bowl under the trap. This will catch the water trapped in the pipes. Unscrew the joints and remove the trap. Don't lose the rubber seal rings that are with them. About a pint of water will come out into the bowl.
Remember not to use the taps or pour water into the sink. It will go all over the floor until you replace the trap.
Clean the trap. Use an old toothbrush and hot water. Put it back together, making sure all the rubber rings are in the right place and the joints are screwed up tight. Run some water and check that there are no leaks from the joints.