A hostel is a form of accomodation unlike hotels or motels in that someone staying at a hostel shares their sleeping space/bathroom space/lounges with others staying there. "Janet Thomas, author of At Home in Hostel Territory says, 'The word 'hostel' does not describe a place; it describes attitude, a philosophy, a coming together of culturally diverse people sharing the wonders, high and low, of the traveling adventure.'" (Source: hostel.com)
In general, hostels are used by young people, and are quite popular in Europe. "Backpackers" tend to take advantage of the very inexpensive rates and cultural interactions that hostels offer. Memberships are available that greatly reduce the cost to stay in hostels; Hostelling International is the largest such organization, although American Youth Hostels, International Youth Hostel Federation, and Youth Hostel Association are also well known. It is also possible to purchase a blank card at a hostel being stayed at for a nominal fee, and then get a stamp at each subsequent establishment. When the card is filled (ie, it has a set number of stamps on it), the bearer can usually pay member rates to stay upon presentation of the card.
If you decide to stay in a hostel, be aware that most require identification of some sort. Passports, licenses, and state/country ID cards are most often accepted. Some hostels require work in the morning, such as harvesting eggs or sweeping floors, but this is not always true. The rule for hostels is that they are all different, and one should (according to the literature available, at least) keep an open mind and be ready to fit in at the hostel where one is staying.