If you paid your mother and you want to claim the child care credit, you must report her social security number and she must pay tax on that amount.
The next question is, did she work at your house or at her own residence?
Ordinarily, if you have a household employee, and you paid that employee more than $2000, then you are supposed to pay payroll taxes (6.2% Social Security and 1.45% Medicare), and you must withhold 7.65% of your employee's wages and pay all that to the IRS. This is reported on Schedule H of Form 1040. You don't have to withhold federal income tax unless your employee asks you to. You would provide Form W-2 to your employee.
Your mother would then report the income on her tax return. If she is living with you at your convenience, then she does not have to report room and board as income. See below for information about taxes and household employees.
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If your mother is providing care at her own residence, then it might be better for her to be self-employed. You would pay her using form 1099-MISC, and she would be able to deduct expenses for food, toys, transportation and the use of her house. She would have to pay 15.3% self employment tax plus income tax on the net income, after expenses. All of this would be reported on Schedule C.
