Why does my dishwasher back up my sink?
It sounds like you have a fairly typical dishwasher set-up whereby it drains into your garbage disposer. If the dishwasher is expelling a larger volume of water than can be drained through the garbage disposer and its drain plumbing, that sounds like you have a blockage.

If you have a two-bowl sink and the bowl with the garbage disposer is the only one that drains slowly, then the blockage is either in the garbage disposer or in the drain pipe that connects the garbage disposer to the drain for the other bowl. This is fairly common and happens when people use their garbage disposer for its intended purpose, and large food chunks or slices get caught in there. You can usually access this pipe fairly easily and clear the blockage. Then you can stop using your garbage disposer and instead compose all the junk you were throwing down the drain. :)

If, on the other hand, the bowl without a garbage disposer also drains slowly when you run the water, then the blockage is downstream of the sink and you may need to call a plumber, or else use a snake yourself to try to clear the blockage.