The Eighth Doctor had tried to take no part in the Time War, except for helping its victims when he could but, in "The Night of the Doctor", he realised that he could no longer stand aside & would need to do things that the Doctor would never do. With the help of the Sisterhood of Karn he regenerated into "a warrior" -- the War Doctor fought. He did not call himself the Doctor ("Doctor no more," as he said). Others -- mainly Time Lords & Daleks -- still referred to him as "the Doctor" but, because he didn't call himself the Doctor, he's not counted as one of the "numbered" incarnations.
Eventually, with the help of the Tenth & Eleventh Doctors, Clara Oswald & the weapon known as the Moment, he brought the Time War to an end in "The Day of the Doctor", then regenerated into the Ninth Doctor.
Because the Doctor believed that, in this incarnation, he had wiped out his own people along with the Daleks, this incarnation was viewed with shame by later incarnations (the Ninth, Tenth and Eleventh Doctors) -- up to the point at which the Eleventh Doctor took part in the events of "The Day of the Doctor". From then on, the Doctor could remember that he had, in fact, saved Gallifrey & its population, while making it appear that they'd been destroyed.
Note: The term "War Doctor" is used out of universe (by fans & so on); he's not called that by any characters in the stories.