Epiphany is celebrated with a variety of customs around the world. In some cultures, the greenery and nativity scenes put up at Christmas are taken down at Epiphany. In other cultures these remain up until February 2. In countries historically shaped by Western Christianity these customs often involve gift giving, "king cakes" and a celebratory close to the Christmas season. In traditionally Orthodox nations, these celebrations typically center around water, baptismal rites and house blessings.
Wikipedia has an informative entry on the holiday if you want to learn more.