Surprisingly, of all the devices that make up an underwater diving kit, the last to be developed was a suitable air reservoir. Demand air valves ( regulators to you and me, ) masks, fins, snorkels, weights, even drysuits may be found in the historical record up to several hundred years ago, but because of material and manufacturing requirements, the high-capacity high-pressure portable air cylinder is a relatively recent development. ( Jacques Cousteau often gets credit for something he did not invent - the demand-valve regulator predates him considerably, although he did much to perfect its application to diving. )
Litigation has caused manufacturers to leave the market, and be replaced by others, who then left, and then others come back. From year to year, there's no telling what brand of tanks may be available, but the specs and recommendations below should apply to any.
Your choice of tanks, in terms of size, number, and mounting hardware, should be closely related to your choice of BC and weight system, so if you have not already, you should take a look at that section.
Air is like money - it's no big deal, until you run out !
Tanks are the most commonly rented of all scuba equipment, and it is often not much more expensive to rent full tanks than to have your own filled. For this reason, tank purchase can often be delayed to the end of your equipment buying period, especially if you are an infrequent diver. Returning rental tanks by the due date can be a pain in the butt though, so you might want to get your own anyway.
Scuba diving cylinders must undergo several types of periodic inspections. The annual visual inspection, or "VIP", is not a legal requirement, but is a standard maintained by the dive industry. Without a current VIP sticker from an accepted source ( such as your local dive shop, ) no respectable operator will fill your tanks, although it would be perfectly legal for you to fill them yourself if you have your own compressor.
Every five years, a tank must undergo a DOT-mandated "hydro" test. This results not in a sticker, but in a marking being permanently stamped on the tank. If a tank fails a VIP test, it may still be salvaged ( for example, internal corrosion may be correctable, ) but when it finally fails hydro, it is condemned - scrap metal - they usually drill a hole in it to make the determination final. The hydro test involves pressurizing the tank to 5/3 of its working pressure with water ( not air. ) Since water is incompressible, if the tank fails, the water will quietly leak out. If air were used, a rupturing tank would destroy a large part of the building it was in.
Much like the annual servicing of regulators, the industry-sponsored VIP inspection is more of a profit center than a safety enhancement. It depends entirely on the person doing the inspection, who may do a good job, or an indifferent job, or a completely incompetent job. Even when done properly, the completeness and effectiveness of a VIP inspection is questionable. The government-required hydro test is much more realistic, but unless you own your own compressor you are not going to get around paying your annual VIP "tank tax."
Scuba tanks are made out of steel or aluminum. Aluminum is cheaper and has somewhat better corrosion resistance, but steel is stronger. The "standard" modern scuba cylinder is the aluminum "80", which actually holds about 77 cubic feet of air at its rated pressure of 3000 psi. This type of tank has probably caught on because for an average-sized person with reasonably good breathing skills, this size is just big enough to cover your allowable no-decompression bottom time and no more. For smaller people, 63 cubic foot aluminum tanks are available, while for larger folks there are choices as big as 120 cuft.
Having said all that about doubles, there are still times when it makes more sense to dive "lite". Out of the water, double tanks are very heavy and cumbersome, which makes them practical only for boat entries, where you simply have to stagger across the deck and fall overboard. At other times, and in less deep situations, you can maintain a safe redundant air supply using a much lighter "pony bottle" - a small tank of 20-40 cubic feet, weighing only about 10-20 lbs, with its own regulator that is used only in an emergency. Some of the situations where this configuration is useful to include medium-depth dives, 40-80 ft, training and practice dives ( especially in the quarry, where it is possible to reach quite deep water from the shore ), and excursions to the tropics.
Larger pony bottles ( 40-50 cf ) are also useful as "swing bottles" for carrying special decompression mixes, while the smaller ones are small enough to actually take with you when you travel. Most airlines will allow it in checked baggage if you remove the valve. This is very useful since many tropical destinations are not equipped to provide the kind of safety equipment that we take for granted here. The smallest pony bottles are really not big enough to get you out of trouble on a deep Jersey dive, however.
Here are some dimensions and specifications for scuba cylinders. This is by no means an exhaustive list. These numbers should be taken as approximate since equivalent tanks from different manufacturers will differ slightly. The two most common models - the Catalina aluminum 80 and Luxfer aluminum 80 - are highlighted, along with my favorite, the OMS LP steel 85.
Weight, full - this is what you will have to be able to lift, in and out of your car, around the dock, and up the boat ladder with all your other gear. Times two for doubles.
Weight, empty - this is pretty much irrelevant
Buoyancy, empty - this is what you need to weight yourself for, so that you can do a safe free stop at 20 feet at the end of your dive
Buoyancy, full - this is what your BC needs to be able to support when you jump in the water, otherwise you are a "dirt dart" heading straight for the bottom and disaster
For their resistance to corrosion, rough handling, and general abuse, and their low price, aluminum tanks are very attractive to rental operations. That doesn't mean you should buy them for yourself.
The proper way to fill a scuba cylinder is very slowly so that the heat generated by the filling process can be bled away as it happens, and the tank does not become overheated. That's a great theory, but in real life, nobody does it that way. Most tanks are "hot-filled", that is to say, that they are filled too fast to prevent them from heating up during the process. If they are filled directly off a compressor, then they will receive a very hot fill. You would be surprised how warm the surface of a tank can get. This is not much of a safety issue, though, as it is well within the thermal limits of the material, either aluminum or steel.
The real issue here is that eventually ( after a few hours ) the tank will cool down, and as it does, the air inside will contract, and the pressure will drop. This typically results in about a 10% loss of pressure. At this point, you could top it off to the rated pressure again and have a truly full tank. Most of the time that's not an option. Therefore, if the shop hot-fills your tank to 3000, you will end up with 2700. To get 3000, you need to fill the tank to about 3300. While this is verboten, it is well within the 5/3 overpressure of the hydro test, and realistically there is no danger. The slight overpressure may wear out your o-rings a little faster, but o-rings are cheap - get a dozen of them at the hardware store for a buck.