Abaddon was known as "the Devourer" for his ability to devour life. His very shadow would kill humans, but his abilities were intended to feed on the Dark. (TV: End of Days, PROSE: The Twilight Streets) Abaddon apparently had no self-control over these powers, and had a limit to how much life he could consume, as Jack Harkness' exposure to Abaddon's life-draining power was fatal for Abaddon due to Jack's immortality. (TV: End of Days)
Abaddon was an ally of the Light, who fought against Pwccm and the Dark. They fought together for centuries, travelling the dimensions until they were caught up in the Cardiff Space-Time Rift. (PROSE: The Twilight Streets)
In 1876, the Cardiff Rift was opened in Tretarri, where Abaddon and Pwccm fought. Though Abaddon was ultimately imprisoned beneath the Rift and appeared to be defeated, in reality he faked the loss so that he could gain power. He also sent his servant, Bilis Manger, to have the event recorded in a special diary, which would be used later to free him. (PROSE: The Twilight Streets)
In 2008, Bilis set in motion a series of events designed to manipulate the Torchwood Three team into using their Rift manipulator to open the Rift and release Abaddon. Bilis' plan succeeded, and Abaddon was freed and rampaged through Cardiff; killing thousands with his shadow, until the immortal Jack Harkness stood in his way. The Demon's shadow eventually killed Jack, and Abaddon died himself in the process due to the excess life from Jack. (TV: End of Days)
Following the defeat of Abaddon, the Light was greatly weakened and was forced to fight against the Dark and Pwccm on their own. After Torchwood sealed the Dark away and sent the Light back into the Rift to protect it, Abaddon's ashes appeared out of the ground and were recovered by Bilis Manger to be buried outside of Britain. (PROSE: The Twilight Streets)
Behind the scenes
The series 1 version of the Torchwood website had a letter home (titled "Aunt Rich's letter home", but signed Aunt Much) in 1827 Mesopotamia to the writer's niece, Lucy. It told of an archaeological dig which was suddenly hit by a "real howler of a sandstorm" that knocked the wall onto several people. As the storm grew and grew, the men at the site awoke suddenly, saying in unison, "He is coming. He will lead us into darkness." After work resumed, George guided Lucy's aunt down to the site of the accident and saw, freshly dug out of the trenches, "a giant carving of a fierce creature, a vengeful demon standing over a powerless city". George identified the creature as Abaddon, and they walked away. A tarpaulin was placed over the sculpture so that the natives wouldn't spot it. 
In Hebrew, the word for "doom" is "avaddon" ("אבדון"). The letter ב in Hebrew can make both a V or B sound.