PADI = Professional Association of Diving Instructors. It is by far the largest of the certifying agencies. A certifying agency is an organization that teaches scuba diving and issues "C-cards", diver certification.
The basic level is usually called open water, and while there are differences between organizations, a C-card from any of them will be accepted by most everyone: shops, dive boats, and so forth.
This Widipekia article gives a good overview of PADI:
http://en.wikipedia.org/wiki/Profess...ng_Instructors