http://www.padi.com/english/default.asp?o=am
Basically PADI = SCUBA Diving in a responsible way - including proper instruction to teach, both classroom for the theory and pool for the practice, then a Check-out exam consists of 4 check out dives for the basic Open Water certification (nicknamed Seacard, or C-Card). There are numerous intermediate and advance courses after the Open Water Certificate.
BTW, SCUBA stands for Self-Contained Underwater Breathing Apparatus. The device was first invented for the Allied in WWII - a product co-invented by en Jacques Cousteau and Emile Gagnan between 1942-43.