In the finale of the first season of the revamped Dr. Who, we see Rose absorb the Time Vortex of the TARDIS and uses this power to not only evaporate the Daleks, but revive Jack.
Due to this process, Jack becomes immortal.
The question is, what was it about this resurrection that allowed him to be immortal rather than just giving him another chance at life?
The problem wasn't so much that she resurrected him, as how she did it. She controlled time itself and wrapped it around him, giving him life with a power that she didn't understand. Imagine it as someone who has a prosthetic limb and needs to learn to control it. She tried to place something (his life) gently back into time and instead punched out violently. Time around him solidified as if it were that paste that goes solid with kinetic energy.
She made it so that his death was something that just couldn't happen, wrapping him in time energy that jumpstarts his system whenever he dies, almost like an antivirus seeking out a problem with a program and fixing it. Of course, over time that energy would wear down and eventually dissipate, in this case by his own choice which allowed him to finally die. Exposure to time travellers would likely drain some of the energy too, as would the vortex he guarded in Torchwood.
It never got explained but I like to think that the whole Face of Boe thing came about due to cellular mutations introduced in his system over the course of millions of reconstitutions over the years. Having the effect turn off (like it did in Miracle Day, for example) every so often would allow him to be mutated from radiation enough that over centuries he could become as he was when he died. That's my personal view on things.
This is answered in Series 3 Episode 11, Utopia - the first episode of the main show in which we find out about Jack's immortality.
Liath's answer is correct, but for one missed point: it was due to Rose's humanity that the Bad Wolf brought Jack back to life permanently rather than just giving him ordinary life.
Jack's life is now a fixed point in space and time:
DOCTOR [behind door]: That's why I left you behind. It's not easy even just looking at you, Jack, because you're wrong.
JACK: Thanks.
DOCTOR [behind door]: You are. I can't help it. I'm a Time Lord. It's instinct. It's in my guts. You're a fixed point in time and space. You're a fact. That's never meant to happen. Even the Tardis reacted against you, tried to shake you off. Flew all the way to the end of the universe just to get rid of you.
JACK: So what you're saying is that you're, er, prejudiced?
DOCTOR [behind door]: I never thought of it like that.
JACK: Shame on you.
DOCTOR [behind door]: Yeah.
And about how he actually came back to life (some less relevant dialogue removed here):
DOCTOR [behind door]: Rose. She came back. Opened the heart of the Tardis and absorbed the time vortex itself. No one's ever mean to have that power. If a Time Lord did that, he'd become a god. A vengeful god. But she was human.
(Flashback to Parting of the Ways, Floor 500 of the satellite.)
ROSE: I bring life.
(Jack remembers his first coming back to life.)
DOCTOR [behind door]: Everything she did was so human. She brought you back to life but she couldn't control it. She brought you back forever. That's something, I suppose. The final act of the Time War was life.
All quotes are from the transcript of Utopia; all emphasis is mine.
My understanding is that Jack's life becomes a fixed point in time and space (much like Pompeii/Mars/Lake Silencio - these are events which will always happen and can never be avoided (we'll ignore the Waters of Mars example for now).
When Rose brought him back she didn't resurrect him she made altered time so that his life is simply fixed. No matter what happens, now much time passes he will always go on.
If you go down the whole Jack/Face of Boe line of thinking things get a little harder to answer!
When Rose looked into the time vortex itself and gained the energy time she had infinite energy and endless power, so when she wrapped this energy around Jack i guess she couldn't controlled it and gave him too much enough to be as you say "Inmortal" however if you remember the episodes of The Face of Boe he says that his energy is running out.In the episode of Torchwood End of Days when the Big demon is about to destroy the world, Jack decides to sacrifice himself by letting the Demon absorb his energy and that the beast won't take it and die. Nearly at the end of the episode Jack survives saying that he was about to ran out of energy.
what was it about this resurrection that allowed him to be immortal rather than just giving him another chance at life?
Let's say he was more than resurrected he was filled with a inmense amount of energy that allowed him to revive everytime he died. However he would die eventually when the energy rans out.
Simple. She brought Jack back to life. But, being linked with time and unable to fully control it, she brought back every Jack. Every single Jack in history was brought back to life.
