The root of the word is the word for light. Epiphaino, in Greek. The same root shows up in words like photograph and phenomenon.
An epiphany is a manifestation and an illumination.
The traditional understanding of the feast of Epiphany is a triple manifestation of the person of Jesus in the gospels: at the nativity, at the baptism in the Jordan, and at the wedding at Cana. I tend to think of it only as Three Kings Day because it closely follows Christmas, but the Roman Catholic readings for the feast take turns going through the stories of the three manifestations through the three year cycle of readings. At the nativity, the child Jesus is made manifest to the shepherds and the magi. When he submits to the baptism of John, Jesus is made manifest as he starts his active teaching ministry. At the wedding at Cana, Jesus is made manifest through the miracle of the water turned to wine -- "This, the first of his signs, Jesus did at Cana in Galilee, and manifested his glory." (John 2:11, ESV)
"Three mysteries mark this holy day: today the star leads the Magi to the infant Christ: today water is changed into wine for the wedding feast: today Christ wills to be baptized by John in the river Jordan to bring us salvation."
(Antiphon for the Magnificat, Evening Prayer II for the feast of the Epiphany, Christian Prayer: The Liturgy of the Hours, Daughters of St. Paul, 1976)
The season of Christmas / Epiphany is great time for Births, Baptisms and Weddings!
[P.S., for extra credit: As a meme, hipsters don't understand "epiphany" any more than they do "irony".]