All (or nearly all) in the movie Fight Club can be explained by the protagonist's mental disorder. But why does he stay alive and get rid of his alter ego after shooting himself? What is the meaning of this?
When you watch the scene closely (I wish I had screenshots...might have to dig later) you can clearly see that he doesn't shoot himself in the back of the head, but really in the side of the mouth, albeit farther back than his cheek.
Even if you feel around with your tongue you can feel that it is just skin and muscle and nothing important until you get to your jawbone, we can assume he hit just before that, which explains why he is still alive. A shot like that would hurt a tremendous amount, and probably bleed a lot, but it won't kill you outright, not unless you don't find proper treatment or something to close the wound. Note I AM NOT A DOCTOR
The rest of this answer is C&P of my answer to a related question
It is then he realizes that he must follow Tyler's own advice and "hit bottom". Only by doing so, is he able to control his own body and "kill" Tyler. To hit bottom he realizes he must do something life-threatening and plain insane.
Whereas the bullet only blew a hole in his cheek, it was his mental F--- You to Tyler, showing him that he can take care of himself now, which is why the bullet is shown to have gone through the back of Tyler's head, killing him.