If you do not already have a show in mind, you can find current listings in the New York Times or online at Broadway.com. It is a good idea not to limit yourself to a single show to keep your seating and pricing options a bit broader. Also, find out what the retail price of the shows are so you know a good deal when you see it.
Once you know what you want to see, the most convenient way to obtain discount tickets is by going to a TKTS location in New York. In addition to their Times Square location, TKTS vendors are located at South Street Seaport and in downtown Brooklyn. Same-day performance tickets to Broadway and off-Broadway shows are sold at TKTS for half of the face value. Day-before tickets are also available for matinee shows, but only at the Brooklyn and Seaport locations.
Broadway shows sell their tickets online, through ticket agencies and at their box offices. Buying directly from the theatre enables you to take advantage of same day discounts and standing room only seats, and save on handling fees associated with outside vendors. Additionally, two hours prior to the show, theatres will make student rush tickets available. With a student identification, when seating is available, "rush" tickets are offered for a considerable discount. Also available for students are box office lotteries where anyone willing to put a little time standing in line has the chance to win big savings. Contact the theatre directly regarding their policy and be prepared to show a current id.
Any city visitor knows there is no one more in-the-know than the concierge at their hotel. Not only can he get his hands on the best shows and the best seats, he can also use his connections to get you into hot restaurants and nightclubs. However, all that pull comes with a steep price tag that doesn't include the tip he will expect from you when the transaction is completed.
In addition to purchasing Broadway tickets through traditional vendors, sites like Stubhub and Craigslist often offer discounts on Broadway shows. The caveat with any online purchase, however, is to be leery of offers that sound too good to be true: they usually are.