We just moved into a new apartment and the kitchen has a double sided sink. The water will not drain out of the sink past the opening of the drain(drain on one side and the disposal on the other). When the disposal is turned on water floods into the other side of the sink and the water level eventually reaches equilibrium at a slightly lower level. If I plug the drain side and run the disposal it will eventually drain beyond being visible in the pipe. However the next time water is run into the sink it reaches equilibrium again.

Will upload a video later today.

You have a blockage in the horizontal drain line in the wall.
The solution would be to remove the "J bend" from the trap and send a snake up into the wall. See the images below, Thanks.

John

I agree with John that you have a blocked drain but before snaking I would try plunging. Block the disposal side. If you don't have the stopper, block with a wet cloth or possibly a large sponge.
Run about an inch of water in the other side and plunge. Plunge rapidly. Note that you get more results from the pulling than the pushing when using a plunger.

I notice that you said "apartment". If plunging doesn't work I suggest that you call the apartment maintenance office.