Dishwasher drain backing up into sink
You have a clog somewhere in your line. The first thing to do is check out your P-Trap because it's fairly easy and doesn't require any tools (assuming the nuts weren't tightened by Hercules). If that doesn't work then it's time for a snake. Snake by removing the trap and going in under the sink. Then you won't have to navigate the P-Trap and will have an easier time.

Look around for cleanouts. If you can't reach the clog from under the sink, moving to a different, downstream access point might bring you close enough to get it.