My mom moved into a Duplex house on July which she paid rent and a deposit of 1200.00 = 2400.00. The owner lost the house in July 16, 2008 but a broker came and talk to my mom and told her that the houses were lost and that the bank was going to be in charge of them. The Broker asked her if she would stay she would pay the bank the same amount of rent as to the owner 1200.00. So my mom sign a contract but it is a monthly contract. But the people in the first house are not paying anything and they said they were going to leave but until this day they are still their , they have been living their for 6 months for free. I don't find it fair but now if my mom wants to leave she still has to pay rent and depending on which week of the month she leaves and it will be decided to get her money back.
MY question was is it true that the bank charges you for staying when the homes are lost and the bank takes over?

You are looking at this from the wrong angle. If you live in a property that someone else owns then you are responsible for paying rent. When the bank foreclosed on the property they became the landlords. This is no different if the previous owner was able to sell the property. They had a choice of evicting the tenants or asking them to stay on and continuing paying rent. Apparently they chose to continue operating the property as a rental.

Since the new lease your mom signed is month to month, the lease can be terminated at any time by either party with sufficient notice.

As for the other tenant, that's between them and the bank. I do not know why the bank hasn't evicted them yet, but eventually they will get around to it. When they do, they will owe for all the unpaid rent and the bank can sue to collect.