At one time Christmas was a twelve-day celebration, hence the song. Christmas Day was the First Day, and the season ended the night of the Twelfth Day, January 5th, which in some countries was an even bigger celebration than December 25th. It was followed by the Epiphany (visit of the Magi) on January 6th.
This is one old tradition that would be nice to have restored. It would help to ease the anticipointment associated with Christmas. (See this strip if you don’t get the reference:)
Happy Second Day of Christmas, everyone.