The Kessel Run was first mentioned in the first Star Wars film that came out in 1997. Obi-Wan Kenobi is trying to get himself, Luke Skywalker and the two droids transport to Alderaan when prospective transport pilot Han Solo brags that his craft, the Millennium Falcon can do the Kessel Run in twelve parsecs. For those that are not aware, a parsec is an unit of length and not time which it was implied it was in the film.
Excluding where it might've been mentioned in the books or cartoons, the Kessel Run is mentioned again in The Force Awakens when Rey is surprised that she's on the Millennium Falcon and has to be corrected by Han Solo when Rey incorrectly describes what the Millennium Falcon is famous for.
It is not until Solo that we learn what the Kessel Run actually is. The planet of Kessel is a planet deep within a nebula of gas, dust, monsters and gravity wells. A gravity well is the pull of gravity that a body in space exerts. In the film, the gravity well looks like a mini black hole which the Millennium Falcon uses to escape a monster in the nebula.
Kessel is controlled by a crime syndicate that uses slaves made up of a variety of races including humans and Wookies to mine Coaxium, a much prized fuel in the galaxy. As the last attempt by Han Solo and Tobias Beckett to get their hands on Coaxium ended in failure, the group had to travel to Kessel in order to steal some unrefined coaxium ore
There is only one safe path from Kessel through the nebula and that is heavily guarded. Han Solo proposes a short cut through the nebula to Savareen where the fuel will be stablized and refined. The Kessel Run is the journey through the nebula to or from the planet.
In World of Warcraft, there is a nod to the Kessel Run on Bloodmyst Isle where you are requested to use an Elekk, a type of elephant to run a certain course round the zone. It isn't the only nod to the Kessel Run but one that springs to mind.