Sometimes you have to unclog a sink. There are a lot of things that you can do to try and free up your drains.

A plunger is not typically thought of when you are not in a toilet, but either a literal plunger or the same type of thing will work well in a lot of situations with not a lot of work. I had a plugged sink in my kitchen and I used a plastic cup like a plunger because I noticed that it was about the same size of the hole. It worked and I haven't had any more stop ups.

Bleach is a great and cheap way to unclog drains, be careful though if it isn't a slow drain then you don't want to put other chemicals in they might mix with the bleach. At around a dollar a gallon it is the cheapest chemical you can use and works almost as well as the drain cleaners.

There are drain cleaners you can buy in the store, don't mix these with bleach or other kinds. I might hesitate a little to use any kind of chemical if the water isn't draining at least a little for this reason.

You can buy a snake reasonably at the hardware store, this will tear through clogs in all kinds of situations. They just aren't as good as what the plumber uses that is motor driven.

Of course you can always call a plumber, but it is expensive. It is expensive because they know what they are doing.