When I think of backpacking, I tend to think of Europe or Australia, but really you can do it anywhere.
Backpacking is really a form of budget travel (as opposed to luxury travel, which I suppose I should also somehow work into my classification system.) Backpacking is also generally a longer-term type of travel than the typical vacation.
Many backpackers are younger, but that's not always true. They carry what they need with them, and usually travel from place to place as cheaply as possible, sometimes booking overnight trains to save money on accommodations, or staying at hostels.
In many places backpacking is a rite of passage. Sometimes recent college grads will spend their last summer traveling this way before entering "the real world." In some countries they have what is called a "Gap Year" where young people travel the world for a year, often before entering college.