-Terry McGinnis is the future son of Warren and Mary McGinnis. Unbeknownst to Terry (or anyone in the current time), Terry is also the genetic son of Bruce Wayne thanks to the machinations of Cadmus and the Batman Beyond project.
-Terry, at an early point in his life, was a member of a fairly violent street gang led by Charlie "Big Time" Bigelow. He was put in juvenile hall for three months, during which time his parents were undergoing a major divorce.
-After a chance encounter with the Jokerz leads Terry to Wayne Manor, Terry discovers the secret of the aging Bruce Wayne: that he is the retired and legendary Batman of the past.
-This meeting happens at the exact same time as Derek Powers, corrupt corporate executive, is having Warren McGinnis murdered. Terry uncovers the truth, and takes the evidence to Bruce.
-When Wayne refuses to be Batman again, Terry steals the Batsuit and, with Wayne's help, brings down Powers' nerve gas production. Afterwards, Bruce decides to take Terry McGinnis under his wing, deciding that the city still needs a Batman to protect it.
-Terry continued in his role as Batman for some time, before an event involving Booster Gold's time travelling suit overloading sent Terry spiralling back to the present, damaging his suit severely in the process.
Terry McGinnis is the Batman of the Future. He's aggressive, he's headstrong, he's straightforward, and he's a little bit of a jerk at times, just like one might expect the Batman of the grim Gotham City to be. While he's starting to level out, Terry is filled with a lot of pent-up aggression, over a number of situations: his parents' divorce, his father's death, and (now) his being flung back in time by a situation not under his control. While he is exceptionally cunning and very very clever - almost as intelligent as Bruce, if not as well-trained or well-honed - he isn't yet capable of wielding that intelligence the way he does his raw anger. Instead of fighting foes with cunning, traps, and perfect plans, Terry tends to rely on his wits; Terry's the sort of person who'll make smartassed remarks in any situation, no matter the trouble it'll get him into, the sort of guy who knows how to make people get angry and lose their cool, and takes frequent advantage of that skill.
That doesn't mean that Terry is nothing but an angry thug, however. Terry has a strong sense of right and wrong behind him; without it, he would never have become Batman in the first place. For Terry, the world isn't a good place, but as long as he's around, he can make things a little better for the people within his power to help. Whether this means standing up to a bully for a helpless kid or beating down a supervillain, Terry is always there to lend a helping hand, though it might be in a gruff and somewhat aggressive manner.
Ultimately, it is his heart, not his strength or his intelligence, that makes Terry the Batman Gotham City needs and deserves. He's a jerk with a heart of gold, and he knows it well. He might have some control issues, he might have a problem with his temper, he might not think before he leaps every time, but his heart is always in the right place, no matter what.