What does "non-reportable" amounts on 1098-T mean for me?

That only means that it is not mandatory that your report the numbers from your 1098-T on your tax return.

The 1098-T is only an informational document. The numbers on it are not required to be entered onto your tax return. However receipt of a 1098-T frequently means you are either eligible for a tuition credit or deduction or possibly your student has taxable scholarship income. 

If you claim the tuition credit, you do need to report that you got one or that you qualify for an exception (the TurboTax interview will handle this)

You claim the tuition credit, or report scholarship income, based on your own financial records, not the 1098-T.

If it does need to be entered and the student is a dependent, the 1098-T is entered on the parents tax return to claim a tuition credit.  It is entered on the student's return if taxable scholarship is being reported.  There are times when it is entered in both places, but then adjustments must be made. 

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There is a tax “loop hole” available. The student reports all his scholarship, up to the amount needed to claim the American Opportunity Credit (AOC), as income on his return. That way, the parents  (or himself, if he is not a dependent) can claim the tuition credit on their return. They can do this because that much tuition was no longer paid by "tax free" scholarship.  You cannot do this if the school’s billing statement specifically shows the scholarships being applied to tuition or if the conditions of the grant are that it be used to pay for qualified expenses.

Using an example: Student has $10,000 in box 5 of the 1098-T and $8000 in box 1. At first glance he/she has $2000 of taxable income and nobody can claim the American opportunity credit. But if she reports $6000 as income on her return, the parents can claim $4000 of qualified expenses on their return.

Books and computers are also qualifying expenses for the AOC. So, extending the example, the student had another $1000 in expenses for those course materials, paid out of pocket, she would only need to report $5000 of taxable scholarship income, instead of $6000.