There are three main types of BC:

Wings

Wings consist of an inflatable bladder worn between the divers back and the cyinder(s). Wings are not a recent development, but have recently become popular again because of technical diving where they are often used, as the technical diver often carries multiple cylinders on his back and/or strapped to the harness's webbing. The bladder and cylinders are fastened to a backplate which is strapped to the diver. The wing design frees the divers sides and front and allows for a large volume bladder with important lift capacity (60 lbs /30 liter Wings are not uncommon). Elasticated webbing around the bladder is used by some to constrict the bladder when not inflated, although there is some dispute as to the wisdom of this addition[2]. Heavy equipment such as diving cylinders can be fixed to or slung from the back plate.
Diver wearing a stabiliser jacket

Stab jacket

Stab jacket, stabiliser jacket, stab, waistcoat, or vest BCs are inflatable vests worn by the diver around the upper torso, and also act as the cylinder harness. The air bladder may extend from the back around the diver's sides ("wraparound") or may only cover the back ("back inflation.") Wraparound bladders are favored by some divers because they may make it easier to maintain balance, both submerged and on the surface. However, some designs have had the tendency to squeeze the diver's torso when inflated. Back inflation (Wing) BC's do not have this problem but have a greater tendency to float the diver face-down on the surface, which presents an obvious hazard in an emergency. Jacket-style BC's typically provide up to 25 litres of buoyancy (depending on size) and are fairly comfortable to wear. Jacket-style BC's are the most common type among recreational divers because they can integrate buoyancy control, weights, attachment points for auxiliary gear, and cylinder retention in a single piece of gear. The diver need only attach a cylinder and regulator set in order to have a complete SCUBA set. Some "tech-rec" jacket-style BC's have the ability to carry multiple cylinders.
Diver wearing an adjustable buoyancy life jacket

Diver's lifejacket

Adjustable Buoyancy Life Jackets, ABLJs, or Horsecollar BCs are worn around the neck and chest with straps around the waist and usually between the legs. They are historically derived from Mae West lifejackets which were cheap and available. As they were developed in the 1960s [3] they have been largely superseded by wing and vest type BCs. Additionally, they are relatively less comfortable because they use a crotch strap between the legs, and generally provide less total buoyancy potential (up to 15 liters) than Jacket BC or Wings, even though this amount is still generally adequate for both warm water and cold water recreational diving. Finally, because the diver must use a separate cylinder harness as a platform for the aqualung, they are slightly more complicated to use. However, its location on the diver's chest means that it does the best job of all BCs in floating a distressed, fatigued and/or unconscious diver face-up on the surface in the event of a problem.