You need to keep track of your own taxes. It's best to stash the cash away in a savings account, and always make sure you have enough set aside to cover your tax burden. You'll likely be paying quarterly state and Federal income taxes.

To determine how much you owe, calculate your Gross Adjusted Income for the quarter or by looking back at last year's tax return, if you had a similar arrangement. You don't need to file a tax form every time you pay. Instead, simply send off the quarterly checks, and then file as usual at the start of the next year. Make sure you pay enough to owe the IRS less than $1000 at the normal tax time to avoid having to pay penalties and interest.

You can look on their website to a find a full 2012 schedule of when quarterly payments are supposed to be made. Normally, you need to pay after the end of each quarter, but it's not exact according to the calendar month. April, June, and October are popular months. Your state will likely follow the Federal schedule to make things easier.

I would ask him or her for a receipt or some kind of statement saying how much he is paying you. I doubt, though, that he will file because he is paying you under the table. Just to be on the safe side, you may want to keep track yourself if he isn't willing to give you something that proves how much he is paying you.

Getting paid under the table means:
you don't qualify for unemployment if you stop working
your income isn't "earned" for purposes of some generous tax credits
you don't get disability or workers comp if you're injured on the job
you don't accumulate anything towards social security (which is disability as well as retirement)

You have the option of legitimizing this income, by calling it "self-employment" and reporting it on Schedule C, but then you pay all of the FICA and Medicare as well as the income tax, and that can be a rather daunting amount of money.