Ok so, if this person works in your home on a schedule that you dictate, you have a household employee. That would be true of a maid, butler, gardner, etc. It is not specific to child care.
Depending on the amount of wages, you will be required to withhold 7.65% of their salary for social security and medicare tax, and you have to forward this tax to the IRS as part of your tax return, along with a 7.65% match from your own income, like any other employer/employee has to withhold and pay social security and medicare tax. You can optionally withhold federal and state income tax, in which case you have to remit that probably quarterly, or you can leave that up to the employee. You will be required to issue the employee a W-2 no later than January 31 of each year. I believe that Turbotax Home & Business can issue the W-2 electronically, and there are also private web sites that will e-file W-2s and 1099s for a small fee.
It's possible that you won't owe social security tax for the remainder of 2016, depending on the person's wages, but you still have to issue a W-2, and for 2017 you will probably need to withhold and pay social security tax.
Then for the dependent care credit, you already have the employee's SSN in order to file their W-2, so you use the SSN as the provider on your form 2441 and take the credit for your qualifying expenses.
If you decide to pay the person "under the table" and not issue them a W-2 and not report their income or pay taxes, you also can't claim the credit, because claiming the credit will cause the IRS to be on the lookout for matching income from that person.
https://www.irs.gov/businesses/small-businesses-self-employed/hiring-household-employees
https://www.irs.gov/pub/irs-pdf/p926.pdf
https://www.irs.gov/taxtopics/tc756.html
