I unscrewed the nut to release the trap, hoping to unclog my drain. I noticed that there wasn't much hair in there. Suddenly tons of water started coming out into a stew pot I had underneath and almost filled it. I put the nut back on and all this soapy water started coming up into my sink from drain. I could hear the water from my neighbor upstairs. I think they were doing laundry. I am not having any other problems in my unit. Toilet flushing fine, Kitchen sink is draining fine, tub and shower draining fine. When I use the water, it doesn't effect my bathroom sink. Only when I hear my neighbor turn on, I think the washing machine.
What do you think?

I think you need to report this immediately to your apartment manager or apartment owner. This could become a serious flood in your apartment if you are not home to keep putting pots under the drain.

You can certainly report this to the manager but its likely not really much of an issue if it isn't happening with the trap in place. Washing machine pump out very fast and can over tax a drain line that's not 100% clear, your trap will not only stop sewer gas but some small back flows.

Twink's correct! This should be reported ASAP to management. You have a blockage downstream from both the upper washer and your bathroom lavatory. Since this blockage is in a common area there should be no charge to you. Good luck, Tom
Hey Bob! Posted right on top of you. Great talking with you last week

Thanks for the advice! Your right I am currently being held hostage in my apartment just waiting to hear water so I can cup out the water from my sink into my toilet. I feel like I'm in a sinking boat. Was hoping I could handle this myself, oh well. Glad to know that this is clearly a management job! Does this involve a snake in my bathroom drain or is this in the main sewer? Thanks again :)

Contact your upstairs neighbor and let him know his washer's backing up into your lavatory. Not knowing where the cleanouts are I can't say where it will be snaked from. Have you contacted management yet? Tom

Yes I just called the owner. He is calling my neighbors to let them know not to use their washer. He said the plumber should be here in next couple of hours. Thanks