The Yeti is a monster who worked at Monsters, Inc. until he got banished (presumably by Waternoose for tampering with mail) to the Himalayas. Despite being banished, he likes living in the human world.
The Yeti is first mentioned by Mike along with Loch Ness and Bigfoot as the three monsters to all have one thing in common: banishment. When Mike and Sulley get banished to the wasteland in the Himalayas, they fought each other until the shadow of a huge and seemingly menacing figure appears. It shortly turned out to be the Yeti, who greets them and makes them feel comfortable in his cave while offering yellow snow cones (which he claims to be lemon-flavored while Mike thought it were made from urine). He also shows himself to be a comedian, such as when he mentions about his friend Bigfoot, who has called himself "King Itchy" to cheer up Sulley, who still felt guilty about scaring Boo, the human girl that he has accidentally brought into the Monster World, during the scare simulator demonstration. When he mentions something about a local village, Sulley, still wanting to help Boo, suddenly demands the Yeti to tell if there are any kids in that village (which he did, saying that there are all kinds of kids there and that it takes a 3-day trip down the mountain to get to the village) so that he will be able to sneak back into the Monster World. However, the Yeti warns Sulley against going out in a blizzard, but Sulley does so anyway, saying that they need to get to Boo. When he got hit by a snow cone, the Yeti points out that Mike threw it, who throws the rest at him while berating him of what he did and how it will affect his record, though he dropped the last one when Sulley told him that the last part didn't matter anymore. Wanting to exit the heated argument and noticing that they were out of snow cones, the Yeti steps out to "make some more".
The Yeti is last seen carrying more snow cones (apparently what he said wasn't just an excuse) when Sulley slides past him on a sled. The Yeti calls out to Sulley that he has made more snow cones, but Sulley's already long gone and makes it back to Monstropolis to save Boo. Mike eventually takes two of his snow cones and follows Sulley to Randall’s lab where he throws one at Sulley then the other one at him (which unintentionally hits Randall instead, making him visible so Sulley could punch him to stop him from choking him to death).
The 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, implying this was the reason he was banished later on.
The Yeti made a cameo appearance in Cars as a snowplow during the end credits, as part of a meta-joke about John Ratzenberger appearing in every Pixar movie as pointed out by Mack (who is also voiced by Ratzenberger).
The Yeti (called the Abominable Snowman), Bigfoot, and the Loch Ness Monster were all mentioned earlier by Mike to have been banished from Monstropolis when he was telling Sulley about what the consequences of their actions for helping Boo might be, possibly referencing the supposed existence of the real-life versions of said monsters.
It is revealed in Monsters University that the 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 an homage to his voice actor's role as Cliff Clavin, the mailman in Cheers.
Despite previously working with Mike and Sulley, they do not seem to recognize each other in Monsters Inc. Their entire conversation seems to suggest that Yeti is just meeting Mike and Sulley for the first time. Though, since Mike mentioned him earlier in Monsters, Inc., it's likely they still do know each other but just decided not to point it out.
In an M.I. book, Mike, Sulley, Boo, and Celia take the Yeti back to Monstropolis. However, he decides to stay in the human world, anyway.
The Abominable Snowman was planned to appear in the first Disney Infinity game but got scrapped for unknown reasons.