The government/military was conducting experiments on test subjects in order to create superhumans with psychokinetic abilities. I guess the purpose may have been to use these superpeople as weapons in warfare, but that's a bit besides the point. The most promising of these initial test subjects was Akira, a kid with super potential. I believe he became too powerful and could not contain his power and thus created an explosion that took most of Tokyo with it and also started WWIII. After the explosion, scientists were still able to collect parts of Akira's body which they froze and keep in a secret lab under the Neo Tokyo Stadium.

Akira was not the only person with abilities. The little blue guys also have powers, but are not as powerful. An antigovernment group tries to help one of them escape their home/prison when Tetsuo crashes with him. For some reason, perhaps because of his exposure to the blue guy, Tetsuo is brought to the place where they conduct this research rather than a normal hospital. During a scan, they discover that Tetsuo has potential for psychokinetic powers and so they awaken those powers and put him on a superdrug they've concocted which is supposed to help him keep the powers under control.

Anyways, at the core of the story, Tetsuo escapes from his government captors and without the drug, can't contain his powers which makes him grow more powerful, but eventually causes him to grow uncontrollably .. until finally causing an explosion similar to the one created by Akira some 30 years earlier, essentially repeating history.

There's a lot more details, particularly in the manga (f.ex Akira can still project his image and stuff from his frozen tubes and so is a conscious force appearing in the film and even more in the manga), but that's the jist of it I reckon