First of all look at the documents you signed when you moved in. Those will dictate what you must do at the end of the lease period. They will tell you when you must tell them that you want to extend, and when they have to tell you if you can extend. Sometimes they ask you to sign a new lease to extend, other times they convert you to a month-to-month lease. Even a month-to-month lease has rules regarding notice. All of this is in the lease.
If they are expecting two months notice, that sounds like they thought you were staying longer. Did you tell them you were staying longer?
What you need to know is what automatic provisions kicked in regarding the end of the lease.
Regarding the money. You pay the rent in advance. That means that when you move out you don't owe them any more money. In fact they are still holding your security deposit, which they will only return after they inspect the apartment, and you return the keys and parking pass. Note the security deposit is not the same thing as last months rent.
If you move out owing money they can take you to court. Depending on the amount of rent owed, and the condition of the apartment, it will likely be in small claims court. Also keep in mind that if you skip out you won't be there for the inspection, which means they will be the one documenting the condition of the unit.
Check with the state/county/city for information regarding tenant/landlord law. for a website/pamphlet that explains much of this.