A term common in sci-fi, fantasy and other fiction fandoms, a "universe" is the world, environment, and history where a number of stories takes place in the same continuity.
For instance, a comic book publisher may have a number of their titles occupying the same universe, meaning they share the same past and events occurring in one title can affect the characters in others. A popular novelist may establish a universe for several of their books, or give permission for other writers to publish stories taking place in that same universe. A series of TV shows may be set before, after, or in parallel to each other, even to the point of characters from one show making major appearances in another.
The key idea of a "universe" is that it's unified -- the timeline and history are established and the characters are consistent. No matter how many writers contribute how many stories, they can all be reconciled with each other. The result is a much greater degree of immersion for the reader or viewer and an established link with fans of the exisiting stories. The writer's job isn't necessarily easier for using an existing universe, however -- more often than not, fans turn out to have a better knowledge of the universe's history than the writer and complain loudly when inconsistencies are introduced.
Similarly, an alternate universe describes a fictional universe which shares the same history, the same characters, or both with another established universe, but diverges from it at some fixed point. From then on, the two universes are no longer consistent with each other. Most unauthorized fan fiction exists in alternate universes.
See also: parallel universe, multiverse, Expanded Universe: More Worlds of Robert Heinlein, Known Space Universe