Loosen and remove the couplings that hold the trap to the straight pipes that run from the sink and to the sewer. You may need to use a plumber’s wrench. If the pipe is plugged, all kinds of messy stuff will fall into the bucket and must be removed from the trap. If possible, take the trap outside and shoot a stream of water from the hose through it. Rinse it out thoroughly. Use a snake or wire from a coat hanger to remove built-up debris from the pipes.
Also, pull out the sink’s drain plug to clean it. The plug pops up and down in the sink. It is attached to a rod and lever that are held to the back of the drain pipe by a nut. Pull out the rod by unscrewing the nut. Clean the drain plug and remove any gunk that’s in the pipe, then reassemble the plug. Reassemble the trap, tightening the couplings by hand so you don’t screw them too tight. Flush the drain by running hot water for a few minutes.