Michael Jon Carter was a star athlete of the 25th century, until a gambling scandal cost him his reputation. As a night watchman of the Metropolis Space Museum, he was enamored by 20th century heroes and so stole a time sphere to travel back to that era with a Legion of Super-Heroes flight ring and force field belt, wrist blasters and other weaponry. As Booster Gold, he attempted to make a name for himself and brand his identity as a corporation, before truly learning how to be an actual superhero after meeting Superman.
Congressman Thomas K. Cole was an aspiring politician in America during World War II, and his zeal led to a conflict with Earth-Two's Wonder Woman. When a foe of hers witnessed this animosity between the two, he was determined to make Cole his pawn by endowing him with superhuman powers of fight, energy blasts and a durable physical frame through an amulet Cole wore. As the Bombardier, he would aid the Amazing Amazon against her foes Paula Von Gunther and the Red Panzer as well as their Nazi German force. Then, when he was revealed to be the true threat, Wonder Woman destroyed his amulet reverting him back to normal.
While Booster Gold had a career in a potential future on Earth-One, as evidence by scenes of him with his robotic assistant Skeets in the Space Museum and later fighting alongside that world's Thorn, in that timeline his career was mostly not chronicled. It would become more prominent in a merged Earth after the Crisis on Infinite Earths. As for Bombardier, after his own alien artifact was demolished there is no documented evidence that he returned as a masked man on Earth-two.