I have a dual sink in the kitchen and for some reason the water will not drain. Even if I turn on the garbage disposal, I hear the noise but nothing happens. Actually, the water on one side of the sink actually rises! Help

Actually, the water on one side of the sink actually rises! That's because the table in the disposal acts like a centrifugal pump.
You have a clog. Most likely In the horizontal drain line in the wall.
To repair remove the bend of the trap and send your snake up into the wall, (see image) Put out about 7 foot of snake and flush the line with hot water. Good luck, Tom