The Yeti is a monster who worked at Monsters, Inc. until he got banished (presumably for tampering with mail) to the Himalayas. Despite being banished, he likes living in the human world.
In Monsters, Inc. when Mike Wazowski and Sulley get banished to the wasteland in the Himalayas, the Yeti greets them and makes them feel comfortable in his cave. He also shows himself to be a comedian, such as when he mentions about his friend Bigfoot, who has called himself "King Itchy." When he mentions something about a local village, Sulley, still wanting to help the human child he has accidentally brought into the Monster World, suddenly demands the Yeti to tell if there are any kids in that village so that he will be able to sneak back into the Monster World. However, the Yeti warns Sulley never to go out in a blizzard, but Sulley does so anyway.
The Yeti is last seen calling to Sulley that he has made more snow cones (after the lemon ones had been used up as a tossing weapon by Mike), but Sulley just ignores him.
Yeti appeared at the end of the prequel Monsters University as a Monsters, Inc. employee who introduces Mike and Sulley (then new arrivals) to their jobs as mail sorters. He tells them not to mess with the mail or peek into them, as such acts are punishable by banishment. Apparently, this was the reason he was banished in the first place.
He appears during the end credits of Cars as a snowplow when Mack is at the drive-in movie theater.
Yeti resembles Bumble the Abominable Snowman from Rudolph the Red-Nosed Reindeer.
The fourth quote Yeti says below was taken from an old jingle for Armour hot dogs.
It's revealed in Monsters University that Yeti was actually once a mail sorter at the Monsters, Inc. factory, but was banished to the Himalayas for accidentally messing up his job by tampering with the mail. This is also a homage to his voice actor's role as Cliff Clavin the mailman on Cheers.