Talia Al Ghul is the daughter of criminal mastermind Ra's Al Ghul as well as a love interest to Batman and mother to their love-child Damian.
Biography
Talia was brought up by her father Ra's Al Ghul to be his second-in-command for the League of Assassins. When Ra's heard of Batman's detective prowess, he resolved to marry Talia to him so that he would have a worthy male heir. To attract his attention, Ra's had his servant Dr. Darrk kidnap Talia. The plan worked and both Talia and Batman were imprisoned together. Whilst initially attracted to her, once Batman finds out about Ra's al Ghul's criminal empire he's forced to treat her like an enemy. The Al Ghul family has since become one of Batman's most complex foes, as they wish to have him as an ally yet are willing to destroy what he wants to protect.
Son of the Demon
While teaming up to bring down a terrorist, Ra's Al Ghul offered Batman Talia's hand in marriage as a token of goodwill. Batman accepted and the two consumated their marriage. Talia was impregnated, but apparently lost the baby due to the stress from a recent terrorist attack. The loss of his unborn child was too much for one who had seen his family die before his eyes, so Batman left Talia. In the epilogue, however, it was revealed that the newborn child was alive and well.
Birth of the Red Hood
Talia later discovered that Batman's former sidekick Jason Todd had been mysteriously resurrected following his death by the Joker's hands. Talia took him in, hoping that Batman would grow to love her more because of it. Talia told Jason that Batman never avenged his death, appealing to the young man's fury. Jason was then trained by several assasins and criminals to help enact his revenge against both Batman and the Joker. When Jason turned 18, he took on the identity of the Red Hood and became a new foe for Batman.
Damian
Talia later confronted Batman with the news that their son was alive and well, introducing him to 10-year-old Damian. After entrusting Damian to his father, Talia instigated a terrorist attack by Deathstroke, only to be stopped by Batman and his son. Talia would later become an enemy to the Bat-family after Damian became Robin. She revealed to Damian that she had cloned him after his apparent betrayal and that Damian's body had been bugged so that Deathstroke could use him like a puppet.
Batman Incorporated
Talia would expand her father's empire, renaming it Leviathan and causing widespread terrorism as opposed to Ra's al Ghul's more focused plots. Talia waged war against Bataman's army, using all manner of criminal activities to weaken him. Talia's clone-son was also genetically modified into a full-grown adult and used as her personal bodyguard, known only as The Heretic. The Heretic killed Damian in battle before dying by Talia's hand. Talia then meets her end at the hands of the former Batwoman, Kathy Kane.
In Other Media
Talia has appeared in any media involving Batman and Ra's Al Ghul, such as Batman: The Animated Series and Batman: The Brave and The Bold.
The Dark Knight Rises
Talia Al Ghul arrives in Gotham years after the death of her father Ra's Al Ghul. She goes by the alias Miranda Tate and manages to earn the trust of Bruce Wayne. Their relationship later turned to romance before Bane takes over Gotham. After Batman defeats Bane, she reveals herself to him, stabbing Batman in the side.
It is then revealed that Talia was the daughter of Ra's Al Ghul and had been brought up in prison with her mother and Bane. After the prisoners attacked and killed her mother, Bane helped Talia escape, where she found her father. Ra's then rescued Bane as a way of thanks, only to exile him out of spite for allowing his lover to die. Talia then folloed Bane and refused to speak to her father for many years.
Talia prepares to detonate a nuclear bomb in the city, only to be stopped by Batman, Gordon and Catwoman. She is badly injured from the assault and dies cursing him.
Love Rivals
Catwoman
Talia and Catwoman tend to find themselves at odds with each other due to them being the only two women Batman has felt a connection with. As Catwoman has become an increasingly heroic character, Talia has become increasingly villainous, culminating in Batman Inc., where Catwoman is Batman's occasional partner and Talia is the head of a major terrorist organisation.
Talia also has an intense dislike of whoever Bruce Wayne is publicly dating, going as far as to assasinate one that happened to be a double-agent.