Warning: Wikipedia contains spoilers!
The setting is that of Neo-Tokyo, a Tokyo rebuilt (over what is today Tokyo Bay) after World War III destroyed it. As it turns out, World War III was really started by the uncontrolled growth of the superhuman powers of a child named Akira, who was enrolled in a secret government research program. In the story's current time, 40 years after WWIII, a gang of young bikers led by the cocky Kaneda is involved in a fight with a rival gang when the gang's youngest member, Tetsuo, collides with a mysterious child on the highway. This child has escaped from the government psychic research program. Tetsuo finds and eats a pill (dropped by the child in the collision) which awakens Tetsuo's own latent powers, with disastrous consequences both on the personal level, as old interpersonal conflicts with his friends resurface, and on the larger level, as Neo-Tokyo is threatened by another Akira incident.
Akira, like Otomo's other work (such as Domu, A Child's Dream), revolves around the basic idea of humans with superhuman powers, in particular psychokinetic abilities, but much of the story does not focus on these abilities themselves, but rather the people involved, social issues and politics. The social commentary is not particularly deep or philosophical, but rather a wry look at youth alienation, government corruption and inefficiency, and a military grounded in old-fashioned Japanese honor, displeased with the compromises of modern society. As with most science fiction, the issues in the future world are clearly treatments of contemporary social issues.
This classic movie led the way for the growing popularity of anime, and according to many people caused anime to become quite popular in western Europe in the mid-1990s. In North America Akira was the beginning of the current wave of anime fandom and served as inspiration to the Wachowski brothers' The Matrix trilogy motion pictures.