I was paid a few hundred dollars to review a few textbooks. I submitted an invoice in Dec 2016, and received payment in Jan 2017. When I pay taxes on this income, does that count towards 2016 or 2017?
The context of my question is US federal taxes.
If your business is operating on an accrual basis, the income would be counted as occurring in 2016.
If your business is operating on a cash basis, the income would be counted as occurring in 2017.
If you don't know what that means, you are probably using a cash basis for your business, which means income and expenses take place when you actually receive or incur them. According to cash-based principles, if you receive the check in 2017, that is when you report the income.
For more information, see Accounting Methods in Publication 538. In summary, you can choose both as an individual and as a business which accounting method you want to use, and it is not trivial to change it. Cash basis on a calendar year is more or less the default, and is recommended unless you have a specific reason for doing something else.
According to TurboTax:
You report the income in the year you received it...
Therefore, since you received payment in 2017, you will report it on your 2017 tax return.