Your first step is to check if other sinks in your home are draining, if they don't drain your main line is probably clogged and you'll need a plumber.
A clogged kitchen sink usually happens because your garbage disposal is clogged or your drain pipe is clogged.
The easiest and most expensive way to get it cleared is to call a plumber.
Most of the time it's an easy DIY job.
First you need to get all of the water and food waste out of your sink.
If you have a garbage disposal, check to see if it works.
If your disposal is not spinning, start by trying to free the blades.
To prevent serious injury, look under the sink and unplug the disposal, if you can't unplug it you'll need to turn off the breaker, flip the switch on and off to make sure there is no power to the disposal.
Then you can remove the rubber boot in the drain so you can see into the disposal, use a flash light if necessary.
DO NOT put your fingers into the disposal, use tongs or needle nose pliers to reach down and pull items out.
After it's cleared out, try using the end of a wood spoon or end of a broom stick to spin the blades, you may have to work back and forth.
Once you free the blades, replace the rubber boot, get the power on and press the rest button on the outside of the garbage disposal under the sink.
Turn on the water and start the disposal, if the water drains your good to go, if not your drain pipe is clogged.
Next step, you'll have to get all of the water out of your sink again.
If you have a dishwasher, use a clamp or vise-grips to pinch the hose going to the disposal.
Boil a pot of water and pour it down the drain, wait 10 minuets, if you have a duel basin sink put the drain stopper in the other side, add water to the clogged side and try to plunge the clog down the drain.
Boiling hot water will loosen up many clogs.
Next you could use drain cleaning chemicals to try to clear the clog, but I would avoid using them because if they don't work you'll have to take apart the drain apart and risk splashing them on your skin or in your eyes.
You can mix up a less toxic version of a drain cleaning solution with baking soda & vinegar.
Remove any standing water, mix a 1/3 cup of baking soda and a 1/3 cup of vinegar in a large container (it will start to fizz and expand) and pour some down each drain, wait 10 minuets while boiling another pot of water.
Then pour the boiling water down wait a few minuets and try plunging again.
Hopefully this works or you'll need to take apart the P-trap (the U-shaped pipe under your sink) and unclog your pipes.
To take apart the P-trap, You guessed it, you'll have to get all of the water out of your sink again.
Then I would clear out everything from the cabinet under the sink, you'll need room to work and water may splash.
Get a large tub, some towels.
Put the tub under the P-trap.
If your P-trap is plastic you shouldn't need a wrench to unscrew the plastic nuts, it helps if you use a towel over them, they can be hard to grip and feel sharp.
If it's a metal P-trap you'll need a large wrench.
Remove the P-trap (water will drain) and clean it out.
If the clog was not in the P-trap and you have a metal drain snake run it into the pipe as far as you can to clear the drain.
Put the P-trap back together and try running some water.
Check under the sink as your running water for leaks.
If all this work fails your clog is further down the plumbing pipes and you'll need to get a plumber out to discover exactly where the problem is.