The Devil Gene (in Japanese: デビルの血 Debiru no Chi, lit. "Devil's Blood", "Devil Cell", "Devil Factor" or "Devil's Consanguinity") - although referred to as the "Devil's Power" (in Japanese: デビルの力 Debiru no Chikara) in the Tekken 6 Scenario Campaign - is a genetic abnormality isolated in the people of the Mishima bloodline.
While the exact specifications of this gene are never truly explained, it is generally assumed that the gene allows for a person to have a symbiosis or a parasite/host type relationship with a malignant spirit referred to as a "devil" as well as having it (the gene) control the transformations of one to the other. The exact requirements for these transformations are never fully revealed. The spirits can possess the person at any given time during the person's lifespan, even while still in their mother's womb or (apparently) just after death.
The people that currently possess the Devil Gene are, in the order in which it was "revealed", Kazuya Mishima, Jin Kazama, and Azazel. Heihachi Mishima does not seem to possess the Devil Gene like the other members of his family. It was previously believed for a long time that Jinpachi Mishima also possessed the Devil Gene due to his appearance, dialogues and his ending in Tekken 5, but it was revealed by Tekken director Katsuhiro Harada's twitter page that the Tekken project team dared to mislead fans. He also said that the Tekken stories progress in parallel with each character, and that it only has one true canon (story setting).
It is however important to understand that the concept of the Devil Gene was not thought of until Tekken 4, which then somewhat reconnected the previous three canonical games. Prior to this, the concepts of Devil was entirely spiritual. The original concept, before the coming of a "Devil Gene", was not entirely scrapped though, and has in some cases been incorporated into the new continuity.