With examples from Stargate SG-1 and Back to the Future
I realize that there are many definitions and strong opinions of the difference between alternate universe and alternate reality. The following is provided as a basic guideline to help you shape your own opinions.
There are many plot devices in use when creating great fanfiction. Most of them are easily identified. There's the "pretending to be married in order to infiltrate a group" plot or the "stranded on a deserted island/in a cabin/off-world" scenario to help us create new stories. But there is also another plot device that some like to use: Putting our favorite characters into a situation that is similar, yet slightly different - or most commonly known as alternate universe or alternate reality. Which one you use depends on the circumstances.
Both rely on the same concept: Characters living their lives in a way that is different than what we perceive as the norm.
Alternate reality (or parallel universe) refers to characters living on a separate plane of existence. Their lives were shaped by circumstances and decisions made based on their own reality. Their universe can sometimes be very close to our own reality, but there are always major differences.
Stargate plays with alternate reality scenarios quite a bit. In one episode, Daniel Jackson finds a quantum mirror that transports people and things to whatever alternate reality they want. He finds his teammates in this reality, only to realize that they aren't his teammates - and they never were. They are completely different entities, living in a world that resembles ours but isn't quite the same.
What makes this type of scenario an alternate reality is that the characters are not one and the same. In fact, it's quite possible the alternate versions of a single character could co-exist in the same reality (although not for long). Or die without affecting the other. Proof of this is in the Stargate SG-1 episode, "Point of View," when Teal'c kills the parallel universe version of himself, without any consequences.
Basically, it's the same person at the same time in their life, yet living under completely different circumstances and in a reality that is not our own.
Alternate universe, on the other hand, takes the characters we know and love and put them into situations that are completely different from what we consider the norm. In fact, alternate universe plots are so far off, they don't resemble canon at all. These fics include stories of modern day characters in a historical setting or as vampires, pirates and spies (to name a few).
This does not include those stories in which a modern day character is dreaming about living in the old west, or having adventures on the high seas. It's a good way to play with the AU scenarios, but we still have the "canon" version of the character at the beginning and the end of the story.
And while I'm on the subject, there is yet one more type of AU that is used as a plot device.
Alternate timeline is where one version of characters co-exist in a reality similar to the norm (at least as we know it). The characters are living the same lives, yet the paths they take lead to a different outcome, like forks in the road.
Alternate timelines often occur through a warp in the space-time continuum created when a time traveler does something in the past that changes the future for everyone. Because the past was shaped by the decisions mankind has made, things fall apart when someone comes along and alters one little event.
In Stargate SG-1, for instance: The team is stranded in ancient Egypt. The Stargate gets moved to a different location, which means it doesn't get found by Catherine Langford's father. Flash forward to the present, and we see that Sam Carter is not an Air Force officer, Daniel Jackson is recruited only because the original team buried a video recording in a location they knew would be excavated in the present day, and Jack O'Neill is an embittered owner of a boat he rents out for fishermen. Different versions of our favorite characters, yet the same.
Another example of an alternate timeline is the Back to the Future trilogy. Every time Marty McFly goes back in time, he comes back to find his parents are different. In fact, in one instance his father is dead. Every time he alters something in the past, his future changes with it.
What makes the above examples an alternate timeline scenario is that these people are living in the same reality they started out with, only this new reality is based on altered history. It's important to note that going back in time and correcting the difference that generated the alternate timeline is often, but not always, the focus of an alternate-timeline episode/fanfic.
Many TV shows and movies, especially in the SciFi genre, use all three scenarios a lot as it allows them to play with the various sides of a character in a believable situation. What would happen if someone really did go back in time and met their mother? How would Buffy Summers react if she got the opportunity to meet her alternate reality twin? Heh. Enquiring minds want to know!
* Many thanks to green_grrl for the fantastic suggestions she made to help clarify this topic!!