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If there is no rental agreement or lease in place right now between your mother and your sister, then your mother can simply ask her to leave. She may want to give her 5 business days to be gone and can even make this request known in writing if she so chooses. If the sister does not leave, then your mother can have your county sheriff's office remove them from the premises.
If there is a lease, and payment has not been made, she need only give 72 hours written notice to either pay up on rent or leave. If they do not leave, then the sheriff can remove them.
If there is a lease and there is some other breach, like damage to property, then you must give 30 days written notice to get out of the premises.
In the present case, I assume there is no lease. I recommend a written letter telling them they have 5 days (a reasonable amount of time) to get out or the law will be called. If they do not leave, then call the sheriff.
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Just checking in on you to see if you had any additional questions. I want to make sure you are as comfortable as possible moving forward with this important issue. Thanks