My kitchen sick was clogged, so I used liquid plumber and a plunger to unstop. Now after a few moments when I run water in the sink, water overflows from a dishwasher connection. The dishwasher connection is a big u-shaped pipe with a skinny copper pipe sticking out from it. The copper pipe is attached to a flexible skinny pipe.

Oh, and the dishwasher won't drain completely. I don't understand why the dishwasher connection pipe is overflowing. The dishwasher and sink pipes are not connected (at least from what I can see.)

How can I repair?

Most dishwashers are connected to the sink drain and supply too. The washer drain is a black rubber hose and it can connect to a garbage disposal or to the sink drain just above the trap. When you plung a kitchen sink with a dishwasher you can be pushing water in and out of the dish washer drain and not do too much to you clog. Remove the P traps under the sink, careful they will have a pint of probably yuky water in them. Rod into the horizontal pipe sticking out of the wall, that's your drain and it likely does a 90 degree bend just inside the wall. Come back if this hasn't cleared up your issues, the clog can be further down stream.