Los Angeles is a coastal city on the western coast of the North American continent. In today's political conditions, the city is one of the largest North American cities and is located in Southern California, a state of the United States of America.
In the movie it is unknown if all these conditions are still the case, this is never touched on in the movie. However, it is implied that any political recognition or authoritative law that would otherwise be exercised by the United States or California State government is either non-existent or overridden by the authoritative figure of Elysium and the Civil Cooperation Bureau.
Los Angeles is the dominant Earth-based setting in the movie. Most, if not all, of the movie that takes place on Earth, takes place either in or around the city of Los Angeles.