Of course prior to doing any diving, a basic open water certification is needed. Many training agencies offer this class, PADI being the most popular (but not necessarily the best). An open water certification will teach you the most basic aspects of diving. Due to market pressures, this certification has been watered down over the years and now many divers feel that most OW certifications do not prepare a diver for diving. Of course this varies greatly by instructor but very few people would agree that two or three day classes (which are becoming very popular) offer enough time in the pool for a diver to become comfortable in the water. Besides comfort with basic skills and gear, neutral buoyancy should be mastered at this level. That means a diver should be able to use his buoyancy compensator (BC) to become neutral in the water column. This cannot be taught in a 3 day class.
At the OW level, the diver has many restrictions placed upon him. The maximum depth allowed is 60' (it used to be 130' but because of the culling of material from the class, 60' seemed more appropriate; there is talk of making it 40'). The diver is not allowed to enter mandatory decompressions - that means at any time during the dive, he should be able to make a direct ascent to the surface. The diver is also not allowed to enter what is termed an overhead environment (such as a cave or the inside of a wreck).
This isn't required but it is recommended AOW used to be exactly what it says - advanced. Now it's more of OW continued or sometimes a complete joke of a class depending on your instructor. Many people go straight from OW to AOW with very few or no unsupervised dives. This may be a good idea if the OW class was the watered down version most shops offer today. Should a diver find good OW instruction, he should get some dives under his belt before going on AOW.
At the AOW level, I believe the maximum recommended depth is 100', perhaps 130'. Also night diving is generally introduced at this level.
Cavern is where the diver is first introduced to what is called the overhead environment. Several agencies offer a cavern class. Most of these are technical agencies that focus on what is termed technical diving. PADI also offers a cavern class but I believe there are instructors that will not accept the card as a prerequisite for further cave training. The cavern class introduces the diver to the fundamentals of cave diving. These include trim (staying horizontal or slightly feet up in the water), anti-silting techniques (or how to leave the bottom on the bottom), and reel work (all cave divers must have a line to open water - cavern divers must use a reel to lay out their line).
The most important restriction at the cavern level is that the diver must be in the daylight-zone of the cave at all time. The diver must not be more than 200' from the surface. Like with all cave certifications, the cavern diver can only use 1/3rd of his available gas for penetration (1/3rd in, 1/3rd out, 1/3rd for emergencies). The cavern diver is limited to a single tank and a minimum starting visibility of 40' with a maximum depth of 70'.
The gear needed for cavern diving is very similar to OW with a few modifications. First of all, each team must have a primary reel (with around 400' of line) that is used to penetrate the cavern. Also each diver must have a safety reel or spool with around 150' of line used in case of emergencies (lost team member, lost line). The last modification involved streamlining the divers gear to reduce drag in the water and most importantly to reduce the chances of entanglement in the line.
This class builds on what is taught in the cavern class and is often taken immediately after cavern. Besides refining techniques taught in cavern, the intro diver is taught line protocol; shutting down a failed regulator and how to work with the long hose (a 7' hose is required at this level).
At this level, the gear configuration changes significantly. Double tanks may be used or at the very minimum a single tank with two outlets accepting two regulator first stages. Also line arrows are required - these are used to mark the direction of the exit on the permanent lines in the cave. Finally a 7' hose is required for one of the regulators to allow single-file air sharing.
The intro diver is allowed far more freedom than the cavern diver. Instead of being forced to run a primary reel for the entire dive, he is allowed to tie that reel into the permanent line further into the cave. He can follow that permanent line until it ends or until it splits (e.g. forms a T or a Y). He also must exit at his entrance (as opposed to navigating from one entrance to another). When using single tanks, the intro diver may use 1/3rd of his gas. When diving doubles, he is limited to 1/6th. The maximum depth allowed is 100' with a minimum starting visibility of 30'.
In my opinion the intro diver certification is the most dangerous out there. At this point, the diver knows enough to be dangerous. As many divers choose to stay at this level for a long time (or forever), they are often tempted to go beyond their training and start using more that 1/6th of their double tanks, jumping off the main line into offshoot tunnels or even using DPV's (diver propulsion vehicles). Exceeding the limits of their training seems to be all too common among many intro divers. Consequently a lot of accidents happen at this level. There is talk of making this a time-limited certification.
This class takes the training wheels off and was designed to allow a diver to break the completion of their training into two segments. This is a time-limited certification so the diver is forced to complete their cave training. Besides review of skills taught in intro, jumps and gaps are introduced. A jump is running a reel from the middle of a line to another line. A gap is running a reel from the end of one line to the beginning of another line. The maximum depth is 130' with a minimum visibility of 20'
At this level, the apprentice diver is allowed to use 1/3rd of double cylinders. He is allowed simple jumps or gaps (multiple jumps aren't allowed). Traverses (going from one entrance to another) and circuits (taking a different route out of the cave) aren't permitted. Limited decompression is allowed.
Here the gear changes significantly from OW. Double cylinders are required along with three lights and multiple spools/reels.
This class is usually taken with full cave but having a separate level of certification allows an instructor to certify a marginal apprentice/full cave student at this level instead of making him take both over again.
This is the last class in the progression. Circuits and traverses are allowed along with multiple jumps and gaps. Extended staged decompression is permitted (using a different gas for decompressing than the gas on the divers back). The equivalent narcotic depth is 130' (on nitrogen/oxygen mixtures, this is 130' deep; when helium is added to the breathing gas, this can be far deeper). Again, 1/3rd of double cylinders are allowed for penetration. The minimum starting visibility is 10'. It is usually recommended that the full cave diver take a decompression class to gain more in-depth knowledge of staged decompression. This is sometimes taken with the full cave class.