Padi stands for Professional Association of Diving Instructors.
it is the most popular SCUBA diving certification agency. You pass their course and you get a pretty card and are allowed to dive.

Among divers, there's a joke about the real meaning of PADI:

Put Another Dollar In

As a professional association that's what you'd expect...

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(Anyway, PADI is the best diving certification agency for recreational diving, so it's fair to pay for a good service.
And, sigh, I am a PADI member!...)

PADI, while a great association to be certified with, has earned a certain measure of scorn in the diving community, the foremost being the aforementioned "Put Another Dollar In" gag. A great many also feel that PADI is the way to go for amateurs and tourists, but more serious divers might be better off with another certification. Having been certified with both PADI and NAUI, I can safely say that the NAUI open-water course was more stringent and covered more ground (or water, yuk yuk) than the PADI equivalent. I also tend to like NAUI dive tables over PADI, for some reason.

While I was taking an open water diver certification class from a SCUBA Schools International dive shop, the instructor explained that "Put Another Dollar In" referred specifically to PADI's generous dive tables, which allow greater times at depth and lesser decompression periods between dives. Since the tables allow divers to dive more frequently, divers will theoretically pay more money to the divemasters, dive resort, boat operator, etc. to go on additional dives. Given a choice of which dive certification to teach, the theory alleges that resorts will choose PADI because it gets their customers into the water the quickest and maximizes the resort's potential to sell goods and services to the diver. This is particularly troublesome, since you'd think that a course typically taken by tourists and recreational divers would err on the conservative side, but PADI's dive tables alone show that they push the envelope of safety more than their competitors.