You'll probably have encountered it under the name stab jacket, or just "BC".
There are two distinct types of BCD (Not counting the old ABLJ, which some divers still use): The jacket, and the wing.
The jacket is more mainstream, but the wing is becoming ever more popular. Wings put all buoyancy on the diver's back, thus supplying buoyancy more evenly, and are more streamlined. Again, it is common to hear divers say they are "upgrading from a jacket to a wing", but very rare to hear the reverse.
The downside of the wing is that it requires more setting-up and fine-tuning than a stab jacket, and this can lead to divers trying a wing that has not been set up for them and finding it unpleasant to dive with.
It is therefore really only a good idea to buy a wing if (a) you, or someone you know, know how to set it up properly, and (b) you're going to do enough diving to set up and enjoy the improved trim it offers.
If not, a stab jacket will serve you fine.
Some points to bear in mind:
BCDs with an internal bladder and exterior tough fabric are a better bet than a single-layer design.
No matter what the manufacturers say, there is no such thing as a "technical diving jacket", even if it IS black and covered in D-rings. Tech divers use wings.
A 'wing-style jacket' is not a good compromise between a wing and a jacket - it's just a jacket with more things to go wrong.
Avoid the Dacor/Mares Hub like your life depends on it.
The best harness for a wing is a single piece of webbing threaded through a hard backplate that you can easily customize yourself. A harness with clips and stitching everywhere is weaker and almost certainly more expensive.
The claim "You can't have too much lift" is a lie. A BCD that's too big causes increased drag and may trap air pockets. Extreme divers with large twinsets and multiple side-mounted decompression cylinders consider 25kg of lift more than enough. If you need more than them, you've got a serious problem somewhere.
The most common stab jacket type in the UK is the Buddy range, made by AP Valves.
Halcyon and Dive Rite are the top names in wings, but if you're on a budget, AP Valves also manufacture a typically bomb-proof range of wings for considerably less cost than the others.
I use: Modified AP Valves Buddy Tekwing (D-rings removed, no pull-dump on hose), Combro steel backplate, one-piece harness.