Dip out some of the standing water. Leave 4 inches of water in the sink.
Plug the drain over the disposal. Create a vacuum by centering the plunger over the drain in the other sink and pushing down firmly. Push up and down rapidly on the plunger. You should feel the suction. Pull the plunger off the drain. Run the disposal. If the water flows out, you have fixed the problem.
Plunge again, alternating between the garbage disposal side and the other side of the sink. Plug the drain on the side you are not plunging. Pour hot, but not boiling, water into the drain and repeat plunging until the water flows out.
Open the cabinet and look at the drainpipes. If they are white PVC, try to loosen the slip nut. If you can unscrew it, place a bucket under the U-shaped pipe to catch the water. After removing the pipe, look for a blockage in the U. Clean it out and replace.
Insert the plumbing snake in the drainpipe. Push it in little by little by turning the crank clockwise, until the snake is entirely in the pipe. Continue to push it in and crank until you clear the blockage.