BCs basically come in three types. There is the "jacket" type. In this design, the air bladder wraps around the front of the diver. When it is inflated, there is more or less a "ring" of flotation, so the diver tends to float as though in a pool doughnut. These BCs frequently have large pockets. The downsides of them are that they tend to be heavy (simply because there is a lot of BC), there is a lot of BC on the front of the body, and many people don't like the "squeeze" feeling you get when you inflate them fully on the surface. In addition, although they have pockets, they are often not very manageable or useful when you are actually in the water.
Back inflate BCs have the flotation bladder entirely behind the diver. This leaves the front of the body much less cluttered. There is no squeeze of the diver when the bladder is full, and the whole BC is lighter (good for travel). It is POSSIBLE, if the BC is not adjusted well AND you overinflate the bladder, to have a tendency to be pushed forward at the surface. You will hear people bring this up constantly as an objection to back-inflate BCs, but it requires several things to be wrong for it to be a real issue.
Backplate systems are back-inflate BCs that are modular. They consist of a rigid plate, which can be plastic, aluminum, or steel. Woven through slots in the plate is a harness, which can include releases and multiple D-rings, or be as simple as a single piece of 2" webbing. Behind the plate is a flotation bladder, which can be attached to the plate in a variety of ways. The advantages of this type of system is the ability to customize -- customize sizing in how the harness is adjusted, and customize lift by selecting the appropriate bladder.
Jacket and back-inflate BCs can come with integrated weights. This means that pouches are integrated into the design, which can hold a certain amount of the diver's weight. Some people really like this feature, but to me, it has two drawbacks. One is that it makes the rig difficult to move around on land, once the pouches are installed. Another is that the pouches are often difficult to get into the BC once it is placed onto a tank -- so you don't really have the option of moving the rig around before putting the pouches in, especially on a boat. Integrated weight systems have to be evaluated carefully, because they are trying to fill two absolutely contradictory functions -- hold the weights securely (because losing a large amount of weight at depth can be REALLY dangerous) and make them very easy to release (because when you need to drop weight, you need to do it fast). Few designs perform both functions well, a conclusion I have reached after salvaging quite a number of weight pouches from the bottom of local dives sites
It is always best to dive gear you are contemplating purchasing, if that is possible, because what seems to work well on land sometimes isn't quite as nice to dive as you might think. But if you can't dive the BC, at least try it on WITH a tank in place. Make sure it holds the tank quiet and stable on your back, and that you can adjust the straps until it fits fairly tightly around you.