In Star Trek
The most often seen parallel universe is the one from the TOS episode "Mirror, Mirror". In this universe, the Federation is an evil "Terran Empire". Every person in the standard universe had an "evil twin" in the parallel one. This universe later appeared in DS9 and Star Trek: Enterprise.
In various time-travelling incidents, the time line becomes partially altered. History changes, but certain characters still remember the original timeline. In extreme cases, the time line changes so that the people in question would never have been born yet still exist, and are flying a ship that was never built in the new timeline. The most straight-forward explanation for this conundrum is that time-travelling moves those engaging in it to a parallel universe, and the original timeline remains unchanged.
In the TNG episode "Parallels", we observe many such parallel universes where past events occurred slightly differently. We are told that the number of parallel universes is potentially infinite. The existence of these universes lends further credence to the time-travel theory above.
One point of contact between the universes is the Quantum Mirror. In the episode "There But For the Grace of God" , an alternate Daniel Jackson escaped from his Earth, which was being over-run by Goa'uld. In another episode, "Point of View", an alternate Samantha Carter -- who was a civilian scientist married to O'Neill -- escapes the Goa'uld through the Quantum Mirror, but the alternate Carter starts to suffer from "entropic cascade" due to the two Carters being so different.
In 2008, the Atlantis-1 team went aboard a reality-jumping Daedalus, which was going into random realities. They eventually managed to get back to their own reality and escape.
There is no supporting evidence for the existence of parallel universes in the real world. Usually the phrase is used as a figure of speech. For example, someone daydreaming is often said to be off in his own little universe. However, there are certain deluded crackpots who insist that someone using that phrase must be speaking literally. More specifically, George Lucas has stated the Star Wars Extended Universe is a parallel universe and Trektards insist he must be using the Star Trek definition of the term rather than as a metaphor the way a normal person would.