Batman is the name shared by a line of heroes from the DC Generations universe.
Batman I (Bruce Wayne)
Bruce Wayne (1910 - ?) was the first Batman. As a child in 1920, Bruce Wayne assumed the identity of the Flying Fox to help Superboy defeat the Ultra-Humanite. Later that year, Bruce's parents were killed by a mugger. (Unknown to him, they were forewarned of this by Jonathan Kent, but chose not to change their fate so that Batman could be born.) Young Bruce vowed to fight a war against crime. Bruce trained himself in body and mind through adulthood- becoming an Olympic-level athlete, an escape artist, a master of martial arts and disguise, and a supreme detective.
In 1929, Wayne assumed the identity of Robin to help defeat Lex Luthor. He made his debut as Batman in 1939, inspired by a bat costume worn by his father as well as a mundane bat. Batman first teamed up with Superman later that year to battle the Ultra-Humanite (again), and their team-ups became frequent afterwards. After 1940, he took on young Dick Grayson- his ward- as his partner, Robin (giving him the identity he'd used 11 years prior). In 1945, he joined forces with Captain America to defeat the Joker and the Red Skull.
In 1949, Bruce married, and had a son, Bruce Wayne Jr. By 1964, Batman had retired, passing the role to Dick Grayson; thereafter, the illusion was deliberately forged that Batman was always the same person despite it really being different people in the roles.
In 1979, while (unknown to him) his son was mourning tragic events, Bruce Wayne disappeared while searching for Ra's al Ghul- becoming just short of immortal after defeating Ra's, he took over his organization and gradually made it a force for good. In 1999, Bruce Jr. found his father when seeking out Ra's al Ghul, and switched places with him, allowing Bruce Sr. to become Batman again. He continued as Batman as far in the future as 2919.
Batman II (Dick Grayson)
As Batman, Dick Grayson (1930 - 1969) earned a special status with the Gotham City Police Department, and had a romantic relationship with Batgirl. He also took on Bruce Wayne Jr. as Robin, who also served with other young heroes in the Justice League of America. This Batman and Robin discovered and revived a frozen Captain America.
In 1969, Dick was slain by the Joker in an elaborate trap, and Bruce Jr. took over as Batman.
Batman III (Bruce Wayne Jr.)
Bruce Wayne Jr. (1950 - ?) pursued a romantic relationship with Supergirl. They were nearly married in 1979, but she was slain by Joel Kent. Bruce Jr. became a much darker Batman, consumed with rage and guilt, while taking care of Joel Kent's wife (Mei-Lai Kent) and son Clark).
Bruce Jr. eventually came to deal with his rage after the Justice League of America came to stop him, and he nearly killed Green Lantern, in 1986. He took on Clark as Robin, and tried to pass on the mantle of Batman to him in 1997, but Clark became Knightwing instead. Bruce Jr. found his father in 1999, and switched places with him.