The temporal split version of the Eleventh Doctor was a version who was created by the Tenth Doctor's explosive regeneration. Travelling for the majority of this life with Kathy Mills, an eighteen year old student of Elsworth college near Stockbridge, he discovered his own nature as a temporal split only after an encounter with River Song: but more importantly, he discovered that he was destined to become the leader of Faction Paradox, a fate which eventually happened. It took the efforts of the real Eleventh Doctor to stop him, and he was presumed killed.
This version of the Doctor was created by the damage done to the Doctor's TARDIS by his Tenth Incarnation's regeneration. At the height of the damage, the TARDIS and the Doctor split: one heading off with one version of the Eleventh Doctor, one heading towards Leadworth, the other towards Elsworth College.
"Oi! Breath!" - The Doctor, to a corpse.
Once arriving at Elsworth, the Doctor - disorientated and suffering from regenerative trauma - stumbled into the middle of a situation at Elsworth College. Saved from being killed by Kathy Mills (at the cost of another survivor's life) he awoke inside the college. Quickly deducing that the survivors were all in a state of terror, he realised that their enemy was an energy being that lived on fear, and so he let it gorge itself on him, killing it and nearly killing him; had he not been within fifteen hours of his regenerative cycle, he would have died.
Taking Kathy on
The Doctor, as an antidote for the horrors she had witnessed, offered Kathy a place on his TARDIS. she accepted, and they proceeded to travel together.