Following basic certification, many divers immediately move on to an advanced certification level. Especially since some activities (like deep dives) require an advanced card.
Advanced certification has been criticized for being somewhat of a misnomer. In all certification agencies, the advanced level is attained with a relatively small number of dives. For this reason, it has been suggested to rename the advanced open water level to something like, "Open Water II" (as it was previously called by NAUI) or "Comprehensive Open Water", to better indicate the nature of the course. Regardless of what it is called, those that dive regularly will want to complete the course at some point.
Advanced certification is built around specialities. The idea is that exposure to many different varieties of diving increases your experience level and encourages further development. How much exposure is required differs between agencies. I'll cover the four largest.
PADI requires five "adventure" (i.e., specialty) dives to qualify for Advanced Open Water (AOW) certification. One of these dives must be an Underwater Navigation specialty and another the Deep Diving specialty. The other three are the diver's choice. (Note: Until 2006, night diving was required.)
Unlike other organizations, PADI does not require a minimum number of dives for certification. This means that a diver could be Advanced certified with only 9 dives (4 open water dives + 5 adventure dives), all of which are instructor certified. This makes PADI's AOW certification the quickest, and the easiest to criticize.
NAUI calls their second certification level, "Advanced Scuba Diver", with requirements similar to PADI's. However, NAUI requires six open water dives to qualify for certification. One must be navigation, another must be deep diving, and a third is required to be night or low visibility diving. The other three are chosen at the diver's discretion.
Also like PADI, there is no minimum number of dives required. With only one additional dive required, NAUI Advanced certification is almost as quick as PADI.
SDI prides itself on its Advanced Diver Development Program having higher requirements and demanding more experience before certification. As opposed to just a certain number of speciality dives, SDI requires candidates to complete four specialty courses, with no requirements on the type of specialties. Each specialty requires about two dives, so eight dives total.
Another distinction is that SDI requires a minimum of 25 logged dives to complete certification. Once reached, the scuba diver brings in their dive log and four specialty cards to be awarded Advanced Scuba Diver certification.
SSI's Advanced Open Water Diver certification is more similar to SDI than any other agency. They also require completion of four specialties, with no requirements on the specialty subjects.
24 logged dives are required to complete certification.
This should give you an idea of what's required to complete Advanced certification. The student must decide what specialties they want to explore. In my opinion, the most valuable non-required dive is a buoyancy related specialty. Proper weighting and buoyancy control throughout a dive are valuable skills to master early. Other than that, it should be straight-forward to follow your interests in deciding.
What specialities would you recommend? What do you think about the whole "Advanced Diver" misnomer?