An important and obvious topic, which everyone present is aware of, but which isn't discussed, as such discussion is considered to be uncomfortable.
The expression is of US origin, although the precise source isn't known. The meaning, if not the exact wording, dates from at least the 1950s and is possibly some years older than that. The first reference to the phrase that I have found is in The Charleston Gazette, July 1952:
"Chicago, that's an old Indian word meaning get that elephant out of your room."
It isn't clear quite what the author of that intended, but we can be sure he was being ironic. Chicago is a word coined by the people that now prefer to call themselves Native Americans. Their original meaning for the word isn't known, but we can be sure it wasn't anything to do with elephants.
The first known citation that uses the phrase with the clear intention of conveying our current understanding of the phrase is the title of Typpo and Hastings' book An elephant in the living room: a leader's guide for helping children of alcoholics, 1984.
The number of times that a variety of authors have called on the expression in recent years, whenever a topic that they thought was important and deserved more attention, has caused it to become clichéd. One commonly discussed such topic in the 1980s was used to be called 'the Northern Ireland question' or, more colloquially, the Troubles. The film director Alan Clarke made a documentary called Elephant in 1989. The film's screenplay was written by Bernard MacLaverty, who is reported as previously describing the Troubles as "the elephant in our living room".
More recently, in September 2006, the British artist Banksy set the phrase in visual form with an exhibit of a painted elephant in a room in the Barely Legal exhibition in Los Angeles. The theme of the exhibition was global poverty. By painting the elephant in the same bold pattern as the room's wallpaper, Banksy emphasized the phrase's meaning, by both making the elephant even more obvious and by giving those who chose to ignore it (like the woman in the tableau) an opportunity to pretend that it had blended into the wallpaper background.