The Thunderbolts are typically represented by a team of super-villains acting as heroes. Depending on the incarnation they do so either secretly or publically, willingly or unwillingly.
The original Thunderbolts were the Masters of Evil, led by Baron Zemo, who posed as a new team of heroes after the loss of the Heroes in Onslaught to gain the world's trust and access to government files and information. When Zemo finally made his move, however, the majority of the team sided with humanity and opposed him. They struggled to regain Earth's trust and act as legitimate heroes for several years after that, breaking up several times. The Ogre and Swordsman III were members of this incarnation.
During Civil War, the Thunderbolts were transformed into the official enforcement arm of the Commission on Superhuman Activities, and later became the Colorado branch of the 50 States Initiative, hunting down unregistered superhumans. Taskmaster and Lady Deathstrike were part of the earliest version of this incarnation during Civil War itself, and Swordsman III remained with the group through Secret Invasion.
After Norman Osborn assumed control of HAMMER and the Initiative, he reformed the Thunderbolts into his personal black ops unit and assassination squad. Mister X was part of this incarnation.
Following HAMMER's downfall, the Thunderbolts were reorganized by Steve Rogers, Luke Cage, and Warden John Walker into a kind of work-release program for superhuman prisoners on the Raft. Juggernaut and Hyperion were part of this incarnation.
Later, General "Thunderbolt" Ross reorganized the Thunderbolts under his leadership to pursue private missions. Deadpool was part of this incarnation.
Most of the original Thunderbolts reassembled under the command of Bucky Barnes after the Pleasant Hill fiasco.