I lived in a rented apartment without the landlord knowing. can they sue me for rent money for a time period i was there or any of the utility bills?

Are you then sub-letting from the original tenant? Hard to believe the landlord does not know you are there. Who, if not you, is paying the rent?

With no lease agreement or sub-let agreement and the landlord does not know you are there... the landlord finds out and thinks there is skullduggery afoot, you betcha you can be held responsible for costs.

Did you break the door down while they were on vacation and just live in the house??

It sounds like you are a guest of a tennat that has a lease or rental agreement or you sublease from someone.

If you are just illegally living in someone else's property there are all sorts of things they can do.

Please describe "Adverse Possession". Can a person claim property that you have allowed them to use without a lease after a given period even though I paid all property Taxes during this time? How long must they have used the property before this can happen?

No adverse Possession would not apply to this, they can not be allowed to use, they have to use it without the owners permission, thus the term adverse.

They would have to live on the property without the owners knowledge, they would not be paying anything basically and using it illegally basically for a number of years, ( 15 to 20 normally) but many states no longer has these laws on the books, except for small areas where fences or drive ways are at.