I've heard people talk about buy outs with the landlord. What's this and can it be done after I'm already evicted and owe money? Also, how does it work if there is a different company working at the apartment complex I was evicted from?

A buy out is when you pay a negotiated settlement to get out of a contract early. A lease is a contract. It does not apply to you since you were evicted. Also, the money you owe is probably back rent, since they can't collect rental after eviction.