Most drains use a system called a “trap”. A trap is a piece of pipe that comes apart just below the drain point on your sink. It’s located at a low point in the system so that when you wash heavy items down the drain they often collect here. In a kitchen sink this is where fat can congeal making a plug of solid fat, which can combine with other materials going down the drain causing a clog.
Note: You should avoid putting oils, grease and fat down your drain. If these items do go down your drain make sure to run your garbage disposal with cold water. This will help chop up the fat and allow it to move through your pipes.
If you’re having problems with only one sink, you can try unscrewing the bottom of the trap and try to clear it out. But first make sure your water is turned off and you’re ready with a bucket for any of the nasty stuff that comes out. If this doesn’t solve the problem you probably have a clog elsewhere and need to contact your plumber.