Han's line about the Kessel Run in the movie has a long and convoluted back-story. The various drafts of the film script include notes about Han's boast that the Millennium Falcon is "the ship that made the Kessel Run in less than twelve parsecs". The problem of course is that a parsec is a unit of distance, not time. (At least in English it is - but then the characters are speaking English on the screen, so one may reasonably assume that words have the same meanings they do in regular English.)

The revised fourth draft of the film script makes it very clear what Han's line means. Immediately after the line, it says:

Ben reacts to Solo's stupid attempt to impress them with obvious misinformation.

In other words, Han is trying to boast, and using a word he doesn't know the meaning of, which makes it obvious to Obi-Wan that he's lying through his teeth. Even in the shooting script, Han's line is followed by a note stating that he is "obviously lying".

Which is all well and good, but unfortunately even the acting talent of Sir Alec Guinness did not make it quite clear enough that Han's line was intended to be interpreted as a completely bogus fabrication. The result is that masses of Star Wars fans have tried to come up with a reason why the Kessel Run really could be flown in "less than twelve parsecs".

If you take Han's line at face value (rather than a transparent lie), the most favoured explanation is that the Kessel Run is a smuggling route that skirts a cluster of black holes (known as The Maw). The standard, relatively safe, route around The Maw is somewhat circuitous and measures 18 parsecs long. Daring pilots can cut closer to The Maw, trimming distance off the route, at some risk to life and limb. Han Solo, being the daring pilot he is, managed to cut so close that he got the distance down to less than twelve parsecs, and thereby making the run in record time.

This explanation eventually became so favoured that it has been officially adopted into the Star Wars Expanded Universe canon.

Which raises a whole bunch of other questions about the consistency of what was actually intended versus what has later become canon. In short, a whole Galaxy of woe could have been prevented if only it was made clearer in the first place that Han was simply clueless.

Transcript

Chewbacca: Ah, there you are, my fine friends. Docking Bay 94, this way.
Luke: Why didn't Greedo give us the number?
Chewbacca: Discretion, my good fellow. One cannot be too careful in these parts.
Chewbacca: Case in point: that ludicrous "Greedo" alias our good captain gave you earlier.
Han: Heeeyy! Guess what?
Han: I just found out I was adopted. My real name is "Han Solo".
Chewbacca: <sigh>
Obi-Wan: O-kaay.
Luke: What happened to your accent?
Han: What accent?
[SFX]: < squee fo-bing ping doodle >
R2-D2: That's your ship? It's a piece of junk!
Han: Hey, it made the Kessel Run in less than twelve parsecs.
[SFX]: < po ee-dok-nip bioo >
R2-D2: Twelve parsecs? That's a distance.
Han: Right. Just testing. It's the ship that made the Kessel Run in less than 39.1 light years.