What is a tuition waiver (in the US)?
A tuition waiver is when no tuition charges are paid at all. For instance, at my university, graduate students on research or teaching assistantships are not charged any tuition, nor are their advisors. In this sense, they have been granted a waiver for their tuition.

Note that even partial-year tuition waivers can be significant. For instance, during the 2017-2018 year, MIT would charge over $16,000 for summer tuition for graduate students, unless they're only enrolled in research!

I have yet to see any commentary on the issue of advisor-paid or department-paid tuition counts as a "tuition waiver."