Under the sink is a u-shaped piece of pipe that blocks sewer gas from getting into your home. Unfortunately, it can also block water from going down if too much junk has collected there. This is the infamous “trap”.
To clean it you’ll need a bucket to catch any sewage water that may drip from the pipe, a pair of rubber gloves, an old toothbrush, a damp rag, and some cleanser. If the nuts/clamps are too tight you may also need a large pair of pliers.
Remove the trap and clean it and also clean the exposed pipe ends under the kitchen sink with a damp rag and some cleanser as well.
Replace the sink trap and tighten the nuts and run water down the sink to look for leaks.