Yes, that is called a disk. The disk moves to create a seal with the seat.
The valve disk for a residential plumbing faucet typically being in the shape of a
frustoconical washer composed of a resilient rubber.
Take the whole valve stem with the rubber disk to the hardware store. They will have replacement parts there. Depending on what is available at the store, you can replace just the disk or the whole valve stem.
Edit:
The water leaking past the main shutoff valve(s) into your apartment may be too much for you to be able to repair your valve. In this case, you would talk to your apartment management, they would schedule a professional to replace your main shut-off valve(s), and the building water would be secured and depressurized during this time. The main shut-off valve is the responsibility of your apartment management.
Edit2:
Depending on the faucet style, it could be the seat you are looking at.
http://www.diyadvice.com/diy/plumbing/faucets/two-handle-disk-diaphragm/
There should also be an additional water stop valve under the bathroom sink.
Google.com -
Faucet repair