To say Batman has had a lot of relationships would be an understament. Batman has been with numerous women both in facade and in true feelings. One such woman was Talia Al Ghul. And during one such "interlude" Talia actually took some of Bruce's DNA and basically grew a kid. I'm not joking.
Using science, she accelerated the child's growth and eventually raised him herself with the intent of having him be the true heir to her father Ra's Al Ghul. Things didn't go as planned though, and she was forced to reveal him to Batman...who took it as well as you might think.
Unexpectedly for both parents, Damian took to his fathers way of life as the Batman, and eventually became Robin, though a very vicious and brutal version. He has served as Robin to both Dick Grayson (when he was Batman) and to Bruce himself. And along the way he has tried to redeem himself from the ways of his mother and father.
Damian is quick to judge, and very honest to a fault. He doesn't care about people that much outside of Bruce, Dick, and a few others. However, he's a pet love, and has quite a menagerie in the Batcave.
Damian is the true heir to the Bat mantle, and whether he gets it will depend on whether his headstrong attitude will allow him to live that long.