There is a full range of accommodations that one could utilize on a trip, but for our purposes we will stick to the least expensive. The type of accommodations you choose will depend upon the length of your stay. It is good to keep one simple idea in mind: the cost of your accommodations should have an inverse relationship to the amount of time you plan to travel. (That's for you, econ majors!) In other words, the longer you plan to travel, the less expensive your accommodations should be. (Unless, of course, you're loaded).
That being said, I'm going to assume that if you're traveling for months on end you will probably not want to sleep in a doghouse, and if you're staying only one night you probably won't be considering the Taj Mahal.
Here are some middle of the road options that you may consider. A few are for the more offbeat traveler, while others are more aimed at hotel dwellers.
Hostels
Try to clear your mind of the memories of the horror movie that emerged a few years back with the same name. In reality, hostels are safe places that offer clean and desirable accommodations. These "mini" hotels are generally aimed at young travelers who don't mind sharing a room or bathroom with a few other people. Rates are cheap, and often the experience of meeting with other travelers is worth the lack of luxury you may experience. Many Hostels include a kitchen that allows you to do some of your own cooking, as well as a recreational area for socializing. Of the options available to young travelers with limited funds, Hostels are often the most utilized. For more information, see Hostels.com, Hostelworld.com, or Hostelbookers.com. You'll find reviews that will help you in finding the right place for you.
Couchsurfing
If you're looking for free accommodations and don't mind getting up close and personal with a total stranger, this option may suit you. Couchsurfing.org allows you to email locals in a city about staying at their house/apartment free of charge. The purpose behind the website is to cultivate community among travelers, not to give freeloaders a place to stay. If you use this site, it should be with the intention of giving back in some way whatever you receive from your host, i.e., friendship, conversation, cultural transmission. When searching for a host, be sure to look at their reviews to see what others have had to say. Each host has plenty to say about themselves, so try to find someone you will be compatible with.
Hotels
These are mostly self-explanatory, however Hotelscombined.com allows you to compare many different hotel prices at once. The more you can cut down on research time, the better.
Rentals
Apartment rentals allow you to stay in a furnished place while you are traveling. You'll have access to a kitchen at all times, which may save you money on food. You may also get the luxury of a washer and dryer, as well as multiple rooms.