It is unknown exactly who or what created v-13 (inexplicably pronounced "Nu" in the game), but apparently she was created using the body of Ragna and Jin's supposedly dead sister Saya. v-13 is a sort of mecha-girl who was supposed to be sealed in the Sheol Gate, and was apparently created as an "anti-Sankishin" unit (possibly created to counter Hakumen and other heroes, but at this point it's just speculation), who is programmed to kill anyone she deems as a threat who comes near the Gate (based on the various story arcs, a better question to ask would be who doesn't qualify as hostile to her). When anyone ventures down to the Cauldron in the lowest level of the main Library in Kagutsuchi, v-13 awakens, analyzes the threat, and takes steps to eliminate it.
Normally, v-13 acts as an emotionless killing machine, barely acknowledging anyone beyond her programming and speaking in a machine-like monotone before eventually marking the intruder as a threat and trying to kill them. There are only two known exceptions. In the presence of Ragna, v-13 suddenly turns into a sweet, caring girl with a slightly psychopathic personality. She views Ragna as a potential ally in helping her destroy the world, and there are several story arcs that feature her stabbing Ragna and dragging him with her back into the Cauldron, towards the Sheol Gate. Apparently, v-13 claims to posess a piece of the Azure Grimoire just like Ragna, and believes that by merging with him the two of them can destroy the world with their combined power. The other exception is Noel, where v-13 seems to recognize her as a copy of herself, but can't comprehend why, and therefore decides to destroy her anyway without bothering to find the answer.
In the True Ending, it's revealed that v-13's Azure Grimoire fragment was merely a replica of the real thing, just like the one that Ragna holds. It's also revealed that Noel was actually the 12th (U-12?) version of the "anti-Sankishin unit," and that v-13 happens to be the 13th. This explains why Jin seems to vaguely recognize his sister's face when he meets either one of them, and that they are probably both derived from Saya's original body. When v-13 tries to drag Ragna into the Cauldron, Noel saves him at the last minute, letting v-13 fall by herself and breaking the time loop.
In the plot of Continuum Shift, Sector Seven somehow manages to recover v-13's "soul" and with Kokonoe's help places it into the body of Lambda-11, a slightly obsolete model that Kokonoe reactivated to serve her purposes. In the game, Lambda-11 is a slightly more balanced version of the same character. However v-13 herself was resurrected as a playable character only in Lambda's "Unlimited" mode.
In the story, v-13's personality lies dormant within Lambda until the True Ending, when she takes control and throws her (Lambda's) body in front of a chain to save Ragna's life, then Lambda's body fades away as she gives Ragna the strength to beat both Hazama and Mu-12. However, in the epilogue, Hazama and Relius Clover mention in a dialogue that they found v-13's original body in a surprisingly intact form, foreshadowing a potential appearance by v-13 in the next installment of BlazBlue.
In the game Calamity Trigger, v-13 is a difficult opponent to beat in the hands of a skilled player, mostly because of her Drive that allows her to "blade spam" all over the screen, or summon blades out of thin air and pierce the opponent several times over. However, the downside is that v-13 has weak defensive power, not to mention that the Drive blades have a small minimum range, and a few well-placed hits at close range will bring her down.
Smart v-13 players will spam her Drive and stay away from foes that can only use close range attacks, like Tager or Bang, stabbing them with summoned swords from afar until they die. It is a cheap, but effective tactic.