Eve was the only survivor of Maelon's experiments on Tuchanka in 2185 to cure the Genophage. She, along with a number of other female krogan, were recovered by the salarian Special Tasks Group after it was shut down and transferred to Sur'Kesh.
Her birth name is Urdnot Bakara, but she surrendered it when she became shaman of the female clan. Unlike the trials for males, Bakara was locked in a cave for a week with enough food to last. She managed to tunnel her way to freedom after finding a crystal shard. Mordin names her Eve after she is transferred to the Normandy, saying human mythology seemed appropriate as they are on a human ship.
During the Reaper invasion, the krogan won't secure an alliance with the turians unless a cure for genophage is found and released. Commander Shepard calls a conference to discuss how to save Earth, where they and Primarch Victus convince the salarian Dalatrass to release the female krogan, with the Commander overseeing the transfer.
After arriving at the STG base, the squad comes under attack by Cerberus who was tipped off to the transfer. Battling through Cerberus forces, the squad manages to guide Eve to safety. When she is released, Cerberus attempts one last time to kill her, but she grabs the Urdnot leader's shotgun and kills both Troopers before they can react, saying "[she] can handle [herself]".
Once aboard the Normandy SR-2, she is taken to the Medical Bay for tests while the cure is synthesized. Shepard is free to talk to her during this time as she can provide some interesting commentary about female krogan and Tuchanka's past and hope for its future.
If Wrex is Urdnot clan chief, Eve will tell Shepard that when he's not trying to sire half of Tuchanka, he's the best thing that ever happened to the krogan, but she'd never tell him that. She describes him as a mutant, who's not obsessed with war like many male krogan, which she permits Shepard to pass onto him.
After Shepard arrives on Tuchanka, the krogan prepare to use the Shroud to spread the Genophage cure. At the Hollows, Eve delivers a speech to the gathered krogan clans that they must put aside their differences and fight together against the Reapers. In a joint operation with the turians to attack a Destroyer guarding the site, the krogan lead by Urdnot, leave the camp in Tomkahs. However, the operation goes wrong as the krogan get off to a late start, and the turians end up with a solitary airstrike that goes wrong.
The convoy then comes under attack and Shepard orders them to move out and leave the squad behind. After Shepard manages to fight through the ruins and back to the convoy, Eve proposes an idea to eliminate the Reaper. She mentions that the Shroud is near the two largest Maw Hammers in existence, and they can summon Kalros, "The Mother of all Thresher Maws". Eve says that Tuchanka is as much the krogan's home as it is hers, so if the hammers are activated, then she will come.
As Shepard sets off, Eve heads into a nearby lab with Mordin or Padok Wiks to finish the cure. If Shepard chose to save Maelon's data, then Eve will remain alive after the genophage cure quest line on Tuchanka, and the player will retain a "Krogan Clans" War Asset score of 300 (unless the player sided with the salarians and was forced to kill Wrex; this does not apply if Wreav is the Urdnot leader since Wreav does not discover the betrayal so neither do the krogan).
If the player chose to destroy Maelon's data back in Mass Effect 2, then at the very end of the genophage cure quest line on Tuchanka, regardless of whether or not the player chose to allow Mordin or Wiks to fix the sabotaged cure, Eve will collapse and die due to complications coming from having the genophage cure being based on incomplete data. This will cause several krogan clans to fall into conflict over who was responsible for Eve's death, providing an excuse for many to act on long-standing feuds and reducing the "Krogan Clans". Also, if the quest to disarm the rearmed turian bomb left over from the Krogan Rebellions is left uncompleted, then the bomb detonates and Shepard will receive a message from the Urdnot leader stating that Eve has died.
At the conclusion of the mission, should Shepard cure the Genophage, she states her birth name to the Commander and says that they will always be friends. She later sends a message to Shepard, informing the Commander of the various krogan clans gathering in the Kelphic Valley, whom she promises to speak to and ensure that genophage cure isn't take for granted. Later, before the final assault on Earth, Wrex says he has already impregnated her, though she insists on naming their firstborn "Mordin" in honor of the doctor's sacrifice.
However, if Eve dies at any point, the krogan will hold a funeral for Eve which is attended by many krogan as well as Shepard. Eve's eulogy will be given by the current Urdnot leader.