Wikipedia summarized the legend of Paul's death like this:
The most common belief is that late in the evening on Tuesday, November 8, 1966 (a "stupid, bloody Tuesday"), McCartney, while working on the Sgt. Pepper's Lonely Hearts Club Band album, stormed out of a recording session after an argument with the other Beatles, and rode off in his Austin-Healey. "He didn't notice that the lights had changed" ("A Day in the Life") because he was busy watching the pretty girl on the sidewalk ("Lovely Rita") after narrowly missing her dressed in blue (she's the blur on the back of Abbey Road) jaywalking ("Blue Jay Way"). He then crashed into a light pole (a car crash sound is heard in "Revolution 9") and, dying from massive head injuries, his hair and face burned (having "lost (his) hair" according to "Don't Pass Me By"). He was pronounced dead on a "Wednesday morning at 5 o'clock as the day begins" (the day and time mentioned in "She's Leaving Home"). A funeral procession was held days later (as implied in the Abbey Road album cover), with John presiding over the service and gravedigger George burying the body.According to believers, McCartney had been replaced with the winner of a McCartney look-alike contest. The name of this look-alike has been recorded as William Shears Campbell, Billy Shears (the name of the fictitious leader of Sgt Pepper's Lonely Hearts Club Band - a role actually played by Ringo Starr), William Sheppard (based on the inspiration for the Beatle song "The Continuing Story of Bungalow Bill"), or some combination of the names. According to legend purists, William Shears was given plastic surgery, but they failed to repair a scar on his lip (or it was a result of the surgery), and that's how you can tell the difference between the two Pauls. According to some students of the bass guitar, the style of his playing changed after Sergeant Pepper.
Having died during post-production on Sergeant Pepper, the most clues to his death are said to be found there.
The reality is harder to pin down - was it a publicity stunt by the Beatles or their record company? Overzealous fans with both too much imagination and too much time on their hands? Or did Paul actually die, and these clues point to one of the greatest conspiracies in rock music?
A Detroit disc jockey named Russ Gibb was the first known kickstart of the rumor on a major-market outlet. Several books have been written on the subject (both on Paul's death, and on the hoax about Paul's death), and radio and TV shows have examined the controversy from both sides.
Overall, this can be considered to be one of the major conspiracy theories of the twentieth century. Perhaps without the world-changing implications of the death of, say, a world leader, but a self-propelled behemoth of urban mythology nonetheless.