The Aka-Oni (赤鬼), the red demon, is the best known of all Japanese Oni, and the most misread, because people stop at the crimson skin and the fangs. In folklore, red carries a precise meaning, and a tale taught in every Japanese school changed for good the way this demon is seen. For the general meaning of the Oni (etymology, anatomy, good or evil), see the Oni mask guide; here we stay on red.

What red means
Red belongs to the coloured-Oni system inherited from Buddhism, where each shade matches one of the five hindrances of the mind (the gogai). Red is greed, desire (tonyoku, 貪欲): excessive attachment, emotion felt too strongly. It's no accident that it's also the colour of Yang in Eastern thought, of the sun, fire and summer. The Aka-Oni is not the demon of death, which in Japan is more often tied to white or black, but the demon of raw vitality, of energy overflowing.
The tale every Japanese knows: Naita Aka Oni
There's a story taught in Japanese schools and almost unknown in the West: Naita Aka Oni (泣いた赤鬼, "the red Oni who cried"), written by Hamada Hirosuke in 1933.
A red Oni lives alone in the mountains. He longs to befriend the humans of the village, but they flee at the sight of him. His friend, a blue Oni, offers a plan: the blue will raise havoc in the village, the red will chase him off as a hero, and the humans will finally trust the red. The plan works. The villagers take the red Oni in. But when he goes back to his friend, the blue's house is empty. A letter on the door: if he stayed, the humans would work it out and fear the red again. So he has gone, for good.
This tale lastingly changed how the Aka-Oni is read. Behind the red skin and the fangs there is raw sincerity. The red Oni doesn't scheme or manipulate: what you see is what he is. It's a violent honesty, and that's exactly what makes him endearing. The sacrifice belongs to the blue (see the Ao-Oni).
Red against blue
The red/blue opposition sits at the heart of the Oni imagination. Red is hot and impulsive, it explodes then settles; blue is cold and patient, it lasts. In the tale, red is the naive hero, blue the strategist friend who sacrifices himself. That contrast is why they're so often shown as a pair. In tattooing, red is traditionally framed by flames; for the compositions in detail, see the Oni tattoo article.

FAQ
What does a red Oni mean?
In the Buddhist coloured-Oni system, red matches greed and desire (tonyoku). It's also the colour of Yang: sun, fire, vitality. The Aka-Oni stands for raw energy and unfiltered emotion, not death.
What is the story of the red Oni who cried?
Naita Aka Oni (1933), by Hamada Hirosuke, tells of a red Oni who wants human friendship. His blue friend plays the villain so the red can look like a hero, then disappears so the trick is never found out. The red is accepted but has lost his friend.
What is the difference between the red and blue Oni?
Red is hot and impulsive, tied to desire; blue is cold and calculating, tied to contained anger. In Naita Aka Oni, red is the naive hero and blue the friend who sacrifices himself. Same creature, opposite energies.