It is generally called the 300 AAC Blackout. It is basically 556 cases chopped down to use a larger bullet and a company called AAC came up with it. It is more quiet and has better ballistics than the 5.56 NATO and is supposed to be comparable to the 7.62x39 commie round. It wasn't made to be shot through a suppressor specifically. It can also be used in a regular AR15/M16 mag with a regular BCG as I understand it so if this round became more widely available, our military or even civilians could start using it pretty easily. A lot of other rounds have attempted to fill this role but haven't quite made the grade. 300 Whisper, 6.8, etc...
It is basically a 308 bullet in a modified 5.56 case as stated before. Everything is compatible with a 5.56 AR except for the barrel. Uses same magazines, bolt, etc. It has similar ballistics to the 7.62x39 over long range. Its best found use is a large subsonic round through a short barrel and suppressor. Even if you don't have the suppressor, even a supersonic 147 grain bullet has better impact out of a short barrel AR than the 556. A popular option is to cast 220 grain projectiles and shoot subsonic cast bullets for next to nothing. Since I just finished my AR pistol, finding a 300 blackout 8-9" barrel is next on my list for it.
Cost has always kept me from owning a AR platform till now ive made due with my 5.56 SAIGA and it works well not to start no ar vs ak bs But I will say I do like the modularity of the ar platform re; changing of uppers lowers barrels etc.
The official SAAMI name is 300 AAC Blackout (SAAMI short name is 300 BLK), but 300 AAC, 300 Blackout, 7.62x35 are all fairly common. It is, technically, a SAAMI standardization of the 300-221 (wildcat) and is largely interchangeable with the 300 Whisper (another wildcat cartridge).
The original design goal was to have one cartridge that offered good sub-sonic performance (for use with suppressors) while offering good supersonic terminal performance at short to intermediate ranges, all from relatively (to very) short barrels. The other goal was, as others have pointed out, to have as much commonality of parts with the AR15/M4/M16 platform and parts as possible, so all that is needed to convert a 5.56 rifle is a barrel swap.
It was intended as another, perhaps better, option for the Spec-Ops community. However, it seems to be gaining quite a bit of popularity among civilian shooters, particularly wild boar hunters, mainly because of the growing popularity of suppressor use by civilians. (It's sound performance with a subsonic loads and a suppressor really is very good. )