In literary terms, and generally, an epiphany is a breakthrough or a rare flash of sudden awakening. The word comes from the Greek for "manifestation" or "to show forth." Within Christianity, the 6th of January is the day the divinity of Jesus was revealed to the Magi.
The author James Joyce has been widely credited with the secular spread of the word’s usage in the earlier part of the 20th century, due to his works Stephen Hero and Portrait of the Artist as a Young Man.
It's interesting to note that in French, we don't use this particular word. We just say "révélation", or revelation.