There's actually a bit of confusion about the actual date. It's the 12th Night of Christmas, which appears to originally have been counted from Christmas Eve. So actually the evening of January 5th, the eve of Epiphany (January 6th), is apparently the true 12th night. And it was the final burst of Christmas revelry. Unlike modern times which seem to start Christmas activity around Halloween, Christmas used to be celebrated for 12 days and didn't start until Christmas Day.
However, 12th Night has come to be most celebrated, at least in New Orleans, on January 6th, the 12th day and night after Christmas Day. January 6th is also known as the 12th Day of Christmas, Epiphany or Kings Day. It's the traditional day celebrating the arrival of the Magi or three kings at the manger with their gifts for Jesus.
January 6th by whatever name is marked traditionally with the first King Cake parties, Carnival balls, and other celebrations that starts the Carnival season which is nowadays also called Mardi Gras season. (Mardi Gras means Fat Tuesday and actually just means that day in a strict sense - but that's another story.)