An alternate universe or alternate reality occurs when accepted reality is altered. Characters may actually move into a universe that is separate from their own, they may find themselves in a reality constructed by a powerful being or induced by a drug, or find their own universe altered by time travel.
Episodes Featuring Alternate Universes, Altered Reality or Alternate Timelines
A djinn poisons Dean and traps him in a dream state while it feeds from his blood. The AU is commonly referred to as the wish-verse. It is a construct formed by Dean's subconscious arising from one wish - that his mother had never died.
The Trickster creates a time loop in which Dean dies in every iteration regardless of what Sam does. When the Trickster is discovered by Sam, he ends the loop with Dean dying. Sam spends six months searching for the Trickster. When he finds him, Sam begs him to bring Dean back. The Trickster says he was trying to teach Sam that Dean is his weakness, and his enemies will use this against him, but he does restore the original time line. Dean does not remember his final death, although it is unclear if Sam remembers the previous six months.
Zachariah creates an AU where the Winchesters are Dean Smith and Sam Wesson, two men who are not brothers, and who work for Sandover Bridge & Iron Inc.. They eventually team up to hunt a ghost in the building. At a time when Dean seems to recoil from hunting, Zachariah constructed the reality to demonstrate to Dean that he will always pursue hunting. This reality is simply constructed out of false memories and is not truly alternate per se.
Zachariah creates a future AU where Sam and Dean are no longer hunting together and haven't been for five years. It's the year 2014 and most of the population are either dead or have become zombies due to the Croatoan virus.
The Trickster traps Sam and Dean in a universe based on TV shows and advertisements. The boys finally expose him as the archangel Gabriel, who is trying to teach them to play their roles as Lucifer and Michael's vessels.
Balthazar sends Sam and Dean into an alternate universe, ostensibly to hide the key to where all the angelic weapons are being stored, safe from Raphael. They find themselves in a world where their lives are the basis of a TV show, and they are actually actors - Jared Padalecki and Jensen Ackles. In this universe spells don't work, and angels (and presumably demons) don't exist. When they get back to their own universe, they find out the whole thing was a diversion so that Castiel could access to the weapons.
On Castiel's command, Balthazar goes back in time and stops the Titanic from sinking. This has side effects, changing reality in many ways - Bobby and Ellen are married, Dean drives a mustang, and Cuba is a popular American holiday destination. A confrontation with Fate leads Castiel to reversing the effects, but he leaves Sam and Dean with the memory of what happened.
After being shot by Dick Roman in the head, Bobby must revisit old memories to try and survive. He remembers a hunt with Rufus Turner, his childhood, the last argument he had with his wife Karen, and times with young Dean.
AUs in Fanfiction
In fan fiction, an AU refers to a story that diverges significantly from the canon in the Show. Generally the writer will choose to alter key events in canon (known as the Point of Divergence (POD) ) e.g. John dies when Sam is a baby and the boys are raised by Mary, Dean doesn't go to hell or Castiel falls and becomes human.
Some stories may become AU as new information is later revealed on the Show, this is especially true of stories written between seasons, where a fic is based on certain assumptions of what may occur, that are then disproved by later canon, also known as being Jossed, a term coined in relation to Joss Whedon and the Buffy verse. In Supernatural fandom the term Kripked was coined to describe the opposite phenomenon - when events in fan fiction are later confirmed by canon.
If a story is generally compliant with the major events, the introduction of a sexual relationship between characters - whether it's Wincest, or Dean/Castiel or Gabriel/Balthazar, is not considered to make a story an AU.
Some stories may be set in a totally different universe (e.g. Sam, Dean and Castiel are professional bull riders or FBI agents). These stories more closely resemble original fiction in that the characters share little in common, other than their names and some personality traits, with their canonical versions. These stories are more common in RPF fics. In RPF AU stories are essentially original fiction, where the major characters are named, and occasionally very loosely based, on real people.