Try snaking the drain to possibly grab the clog if it's made by hair or something that can catch on the tool.
If the clog is caused by grease, a grease fighting dish soap and hot water can help dissolve the clog.
Pour 1 cup of baking soda in the drain and top with 1 cup of vinegar. Let sit 5-10 minutes then rinse with hot water.
Place plunger over the drain and cover with some water if there isn't already standing water. This will help create a seal.
If you have a double sink, plug the unclogged side's drain opening before starting so water doesn't come out that side when plunging.
Then plunge gently several times. Don't be too forceful as it can damage some connector pieces.
The clog could be sitting in the elbow. Place a bucket underneath before removing to check for the clog as water will be sitting in it.
If none of these methods work and you tried a chemical cleaner without success (or don't want to use one) the clog may require a professional.