Originally developed as an ersatz Guilty Gear due to the original license being owned by Sammy, the BlazBlue franchise has grown into its own universe with everything from the original fighting games to light novels and even an anime adaptation set to appear with most of the same voice cast from the game.
Many years ago, a large being appeared known only as "The Black Beast." It ravaged the entire world, leaving destruction in its wake. All conventional weapons were useless against it, and even nuclear bombs proved to be useless against this strange new threat. But one day, a legendary band of warriors known collectively as "The Six Heroes," wielding magic and special weaponry, rose to fight the Black Beast and bring peace to what was left of humanity.
In the aftermath, the Novus Orbis Librarium ("The Library" or the NOL) united most of humanity under a single government, and organized the construction of "Hierarchical Cities" at high elevations. The death of the Black Beast covered most of the world in "seithr," a magical energy that could be used to power magic and technology, but at the same time too much exposure could cause living things to go insane or even die, so these cities were built at high elevations, on mountains, and similar formations.
There was a relative period of peace, but soon a faction emerged that sought independence from the rule of the Library. The Federation led by the city of Ikaruga ended up fighting in a civil war with the Library, but the city was mercilessly wiped out by NOL forces, and the survivors scattered to the other cities.
Now in the present, in the 13th Hierarchical City of Kagutsuchi, a notorious SS-class criminal known as "Ragna the Bloodedge" has reportedly set his sights on destroying the main NOL branch there, just as he has in other cities around the world. This has sparked bounty hunters and soldiers of all kinds to flock to the city. Some seek the massive bounty his capture would bring, others seek the "Azure Grimoire" (or BlazBlue) that Ragna supposedly holds, an artifact that contains immense magical power to anyone who wields it, and still others have their own reasons for coming to the city.
Noel Vermillion has been revealed as the true holder of the Azure Grimoire, but Hazama appeared at the last moment to steal it from her. While Ragna and Jin nurse their respective wounds, the major players suddenly take an interest in Noel as Hazama seeks to turn her back into Mu-12. Meanwhile, Rachel is having issues dealing with a highly advanced AI that has been resetting the flow of time over and over, Tsubaki is wrestling with her duty to the Imperator over being ordered to kill her dear friends Noel and Jin, Hakumen is being kept prisoner and Kokonoe prepares her contingency plan should things go wrong.
BlazBlue: Calamity Trigger
The game that started it all. Only ten characters were playable in this version, including the final boss character v-13 (pronounced "Nyu," as in the 13th Greek letter).
BlazBlue: Continuum Shift
This game continued off the story from the first game and added two new playable characters to the mix: Tsubaki Yayoi and Hazama. A third, Mu-12, could be unlocked after beating the Story mode. Also, the burst and guard systems were tweaked a bit.
BlazBlue: Continuum Shift II
Technically a patch for the previous game (and was a downloadable patch for the console versions) but released as a brand new title on the PSP. This version also introduced three more playable characters that could be downloaded for a price or patched in for free to play against online: Makoto Nanaya, Valkenhayn R. Hellsing, and Platinum the Trinity.
BlazBlue: Continuum Shift Extend
This one contained all previous playable characters for no extra charge and added one more: Relius Clover. Technically another expansion pack and patch to BB:CS, but released as its own retail copy instead with a couple new modes attached: Unlimited Mars mode, forcing the player to fight a string of Unlimited-powered AI opponents with no continues, and Abyss mode which added RPG elements to the normal fighting game as the player took one character down a set number of levels depending on selected difficulty.
BlazBlue: Chrono Phantasma
A true sequel to the events of Continuum Shift, with at least 6 new characters rumored to be added to the current roster.