What happens with my late mother’s car?

1) The full remaining balance left on the car became due when the legal owner (your mother, since it was in her name) died. The financing arm, lender, etc. does not have to force you to pay it all off at once--they could continue accepting monthly payments if they want--but they have the right to require payment in full on threat or pain of repossession.
2) Since your mother was on the title and you were not, it belonged to her. With her passing, if she had a will, it goes to whomever she willed it to. If she did not have a will, it does by "intestate succession"--the rules for inherits what when there is no will. In your state, if she was not married when she passed, if you are an only child, you will inherit it; if you had siblings, you and your siblings all inherit equal interests in (shares of) the car.
3) You will need to get the car into your name if you are inheriting. How to do that depends on what is happening with your mother's estate (everything she left behind). A first step, however, is to get appointed as the estate's personal representative, or the person legally authorized to deal with, act on behalf of, etc. the estate. If you are the only child, or at least the only child interested in this role, you should have no problem in getting appointed by the court. Contact the clerk's office of the probate court for instructions and how to apply to be appointed personal representative.