Hostelling International (HI) is the "official" worldwide hostel organization (it used to be called the International Youth Hostel Federation, or IYHF). HI is a not-for-profit organization representing hostelling associations in over 60 countries, with a total of over 4000 hostels. HI hostels maintain a certain level of quality and they are generally clean and safe. And in many places in the world, especially smaller cities, the HI hostel is probably the only hostel in town. (By the way, Hostelz.com isn't affiliated with HI or any other group. We list HI hostels, hostels belonging to other hostelling groups, and independent hostels.)
If you're not a member you can usually stay by paying an extra fee per night, but it's best to purchase a membership card before you leave on your trip. An HI card will also get you discounts on thousands of travel related goods and services around the world.
Your HI membership card can be used at hostels anywhere in the world, but you must purchase a membership card from the HI association of your home country.