<p>Newmassdad is essentially correct.<br>
In the humanities and social sciences, some of the top schools offer a financial aid package to all the students they admit into their Ph.D. programs. Princeton may be the only school that offers full fellowships for 5 years. Others combine outiright fellowships (tuition plus stipend) for the first couple of years, then expect students to be TFs for the rest of their program. After the first two years, tuition is reduced (but so is the amount of the fellowship); for students who are TFs, tuition is usually waived. I believe that in many places, health insurance is covered by the fellowship (at some universities, it may be mandatory).
How many students a TF is expected to teach depends on the institution. Some years ago, sections were capped at 25 at Berkeley, and 18 at Yale and Harvard. A TF probably has to lead 2 or more sections in a year to make ends meet.
This is the basic scenario. In practice. the better endowed schools have a variety of fellowships students can compete for; there are also external fellowships such as Fulbright to conduct fieldwork, summer fellowships and fellowships for writing up the dissertation. When graduate students go on the job market, it is good to be able to point to teaching experience. TFs may therefore seek to lead sections in different courses to build up their cv.
Newmassdad: I recently talked to a recent Ivy humanities Ph.D. who is now teaching at a LAC. He is still reeling under the realization that he will have 100 exams and papers to grade by himself (no TF), from two courses capped at 50 each, on top of preparing lectures. When he was a TF, teaching two sections, he had fewer than 40 papers and exams to grade and no lectures to prepare. Nostalgia for the good ol' days as a starving graduate student has already hit.</p>