Here’s some advice on seeing affordable theatre in New York: find out a show’s rush/discount policies before going, and always carry your student ID when going to the box office.
Student rush tickets demand both some flexibility (because there are no guarantees) and pre-planning. Many Broadway and off-Broadway shows will sell student rush tickets for $20-40 on the day of performance, typically limited to 1 or 2 tickets per student id. Be sure to carefully review the policies before you go: some box offices demand cash only, others let you use a debit/credit card. Rush tickets might be available when the box office opens in the morning, or might not be available until a few hours before the show ( see the Broadway theatre policies here: http://www.playbill.com/celebritybuzz/article/82428-Broadway-Rush-Lottery-and-Standing-Room-Only-Policies), and the off-Broadway offerings, here: http://www.playbill.com/celebritybuzz/article/144419-Off-Broadway-Rush-Standing-Room-and-Inexpensive-Ticket-Policies). Typically you have better luck scoring rush tickets on week-night shows (particularly for Broadway theatre) during the school year.
A few theatres provide discounted tickets throughout the season: Brooklyn Academy of Music (BAM) and the Signature Theatre have $25 tickets for many shows with limited availability, so you are encouraged to book early. Other theatres ask that you join a mailing list/club online for student ticket prices, like Roundabout’s HipTix (http://www.roundabouttheatre.org/Shows-Events/HipTix.aspx). Unfortunately, many of the online/list programs have age restrictions (usually 30 or 35). Keep your eye on the Village Voice, Time Out, and discount mailing programs for upcoming shows and ticket deals.
Even the “hottest tickets” on Broadway, like The Book of Mormon, can be seen on a student budget (with a fair amount of luck, since Book of Mormon relies on a lottery system). You are in the best city for theatre in the entire country, and even on your paltry student budget that makes all your non-PhD-pursuing friends laugh (and then, if they are really good friends, buy you a pity drink—these good friends are the same ones that greatly appreciate it when you offer them the second discounted rush ticket you got, by the way), you can see a lot of amazing performances.