We joined a Classical Conversations community this school year and I agreed to be a tutor. This is a contract position and because it paid over $600 in the calendar year I got a 1099-MISC form. The IRS also gets this form so it's definitely reportable income.
I am not a CPA, but in my former non-mommy life I did prepare taxes. So I'm not a professional anymore - this is my disclaimer. =)
The easiest way to report the income is on Line 21 of Form 1040 "Other income." Just write in 'tutoring' and the amount. This method means you pay tax on the full earned amount.
Less easy, but better IMHO, is to fill out a Schedule C that attaches to Form 1040. If you use a Schedule C you'll be able to take deductions related to your earnings. For example, materials you purchase specifically to tutor, mileage to tutor meetings, tutor training expenses etc.
When all of these deductions are figured in, I actually pay taxes on a very small amount instead of the whole amount that was over $600.
It's important to note that you should NOT deduct expenses that you would have incurred as a parent anyway. For example, I would have purchased the CC Foundations book and the CDs as a parent so I didn't deduct those. I also drove to CC each week, but I would have driven my kids anyway so I didn't deduct mileage for those trips. You only want to deduct things that were exclusive to tutoring.