Alternate realities can cover quite a bit; generally an alternate reality is a setting where various universal laws have been modified to make certain things possible. For example: magic, telepathy, dragons, teleporters, etc.

Most games are set in an alternate reality of some sort, even if the main aspect of the setting is mundane. For example: The Wild West with magic.

Certain aspects of alternate realities (usually magic) have made certain sociological groups (usually conservative Christians) quite concerned. For more information on this complex topic please see "Hang on... aren't role-playing games supposed to be dangerous?".