!!!! CONTAINS SPOILERS FOR DOCTOR WHO AND TORCHWOOD!!!!
I'll try not to put too many in though, so it'll be a bit cryptic in places.
Jack is (probably) not an alien; he is probably a human from an earth colony in the 51st century.
He lived in the Boeshane peninsula, planet unknown, which was at some point attacked by aliens who abducted Jack's brother Gray and killed his father. Jack (or whatever his real name is) was the first person from his community to join the Time Agency, which is where he got his wrist strap device (the Vortex Manipulator), and became a poster boy in his hometown, known as 'The Face of Boe'
While he was working for the Time agency, he woke up one day and found that two years of his memories had been removed, presumably by his employers. This led him to leave the Agency, steal a Chula warship and start a new business conning Time Agents into buying worthless space junk - that's how he met the Ninth Doctor and rose, as he mistook them for Time Agents, and tried to con them into buying an old space ambulance.
After the Doctor had dealt with the problems this caused (see DW series 1, The Empty Child and The Doctor Dances) Jack joined the crew of the Tardis and was eventually killed by Daleks (DW S1, Bad Wolf/The Parting of the Ways) Rose had been sent home for her own safety, but she opened the Tardis, absorbed the Time Vortex and came back to destroy the Daleks. She also resurrected Jack, but she couldn't control the Time Vortex and it made him 'immortal' - although later events seem to show that Jack can 'give away' the Vortex energy inside him (Torchwood Series 1, Ep 2 and DW series 3 Gridlock)
The Vortex energy was killing Rose, so the Doctor took it out of her into himself, which triggered his regeneration from Nine to Ten, but they then left the space station leaving Jack behind.
Jack then used his Vortex manipulator to try to find the Doctor, aiming for the 21st century, but he got it a bit wrong and landed in 1869. The Vortex manipulator's time travel function then failed leaving him stuck in the 19th century. It apparently took hom until the early 1890's to discover he was immortal, and he then lived through the entire 20th century waiting for a version of the Doctor he could contact (ie after he had initially been picked up) During this time he got involved with Torchwood as a freelancer, and eventually took over Cardiff in 2000 - the implication is that he didn't agree with what the Torchwood in London was doing and after that one was destroyed, he 'remade' Torchwood in the Doctor's honour.
Eventually, the Doctor came back to Cardiff to refuel the Tardid at the rift, as Jack had anticipated, and Jack tried to rejoin the Tardis, (end of Torchwood series 1) but the Doctor saw him and dematerialised. (DW s3 Utopia) Jack clung on to the Tardis which tried to escape him and in doing so took them all to the end of the Universe. The Doctor then explained to Jack that he had abandoned him because of a gut reaction to Jack's Time Vortex-induced immortality, telling Jack that he was a fixed point in time, which he and the TARDIS automatically rejected.
When they got back to Earth, Jack decided to rejoin his team at Torchwood, instead of travelling with the Doctor.
The team then consisted of Gwen, Ianto, Tosh and Owen. During S2 of torchwood, Owen was shot, but Jack used a resurrection glove to bring him back to life - they thought temporarily, but he actually stayed 'dead', but animated. He eventually sacrificed himself averting a nuclear catastrophe, and at the same time Tosh was killed by the person responsible for the threat.
Ianto died during Series Three of Torchwood, Children of Earth, of an alien plague.