First of all, there are two major types of hostels.
(1) establishments that provide short term accommodation to travellers or backpackers
(2) establishments that provide longer term accomdation (often to specific classe of clientèle such as nurses, students, drug addicts, arrested persons subsequently bailed to await trial and homeless people where the hostels are sometimes run by Housing Associations and charities)
But in both aspects, a hostel is a budget-oriented place of accomodation where guests can rent a bed, sometimes a bunk bed in a dormitory and shar a bathroom, lounge and sometimes a kitchen.
Rooms can be mixed or singe-sex, although private rooms may also be available. Hostels are generally cheaper for both the operator and the occupant; many hostels employ their long-term residents as desk clerks or housekeeping staff in exchange for free accommodation.
Hostels are different from hotels in that:
(1) Hostels tend to be budget-oriented. Rates are considerably lower, and many hostels have programs to share books, DVDs, and other items
(2) For those who prefer an informal environment, hostels do not usually have the same level of formality as hostels
(3) For those who prefer to socialize with their fellow guests, hostels usually have more common areas and opportunities to socialize
You can usually find these near places where outdoor activities and activities of cultural exchange are popular. That is why hostels for travellers are sometimes called "backpacker's hostels". They are also definitely more seen in European countries, such as Germany.
Yes, anybody can start a hostel; it's just another business to operate.
If you want to start a youth hostel however, you might want to make a connection with Hostelling International (HI), a non-profit organization composed of more than 90 different Youth Hostel associations representing over 4500 Youth Hostels in over 80 countries.