There are several ways to get cheap Broadway tickets, especially if you are flexible about the show you want to see. The tried and true way to get cheap Broadway tickets is by going to the TKTS booth in the middle of Times Square - a classic New York experience that's shared by tourists and locals alike. The majority of shows dump their extra tickets here each day, and you can check www.tdf.org/tkts for info about what's available. Beware - lines are long and you may not get the show you had your heart set on, and you often have to pay cash. There are TKTS locations at the South Street Seaport and downtown Brooklyn, but the Times Square location is unbeatable if you want some local color while you wait.
TKTS is run by the Theater Development Fund, which offers other opportunities for cheap Broadway tickets. If you're a student, a non-profit employee, or fall into one of a dozen other categories, you can go to tdf.org and sign up for membership, which grants you special ticket offers and discounts year-round. They also have a great section of info for novice theatergoers called NYC Theater 101.
If you do have a particular show or shows in mind, another great opportunity to get cheap Broadway tickets is to call the individual theater box offices a day in advance and ask if they have rush tickets, lotteries, or standing room.
Rush tickets are often (but not always) reserved for students; each night, the theater sets aside some hard-to-sell seats (like the very front row) and offers them at a particular time (usually first thing in the morning, or at 7pm) at a deep discount, sometimes for as little as $20. Be sure to ask if you have to be a student, and check where the seats are located, before coming down and waiting in line at the theater.
Some shows run a lottery each day, where a block of seats are set aside at a discount for people who come by in the morning and enter a drawing. Generally, you have to return at 5pm or so for the drawing, and if you snooze you lose. This option is often cash-only, so come with some Benjamins.
For the young at heart and strong of leg, standing room is another great option for cheap Broadway tickets. Many theaters have a bar across the back of the orchestra seats where audience members can lean and watch the show. These spots can go for as little as $20. Be sure to consider the length of the show; when you call the box office the day before, be sure to ask if its a one-act play or a four-hour megamusical. Bear in mind that not every theater has standing room, nor does every theater sell rush tickets or run a lottery. Calling in advance is key.
A good tip to figure out if your show(s) of choice might be available is to check the weekly box office income at http://broadwayworld.com/grosses.cfm. Look at the column called "Gross % potential" - this basically tells you how much of the theater is actually selling per week. "Wicked" and "Jersey Boys" are almost always at 100% or more, whereas lesser known shows can run at as little as 30% capacity - meaning they almost certainly send inventory to the TKTS booth, or make rush sections available.
Finally, when looking for cheap Broadway tickets, consider that Broadway theater is incredibly diverse, and a show you've never even heard of might be just as exciting and enjoyable as The Lion King. And if you're only spending $20 on a standing room ticket, instead of $140 for a good seat, you are less likely to feel cheated when the show is a stinker (as some of them are, including the famous ones). So use these resources, and have a great time!