I was taught that the 12 Days of Christmas take place Dec 25 through January 6th, the 6th being variously called Epiphany, 12th Night, the Feast of the Three Kings or Little Christmas.
When my sons were young we had a Nativity scene set up and the Three Kings were placed far away and moved every night a bit closer to the "Crib," arriving on the morning of Jan 6th (via Mom magic) That day the boys each received a present brought by the Kings. We had a "favorite" dinner and a special dessert.
This gift aspect very handy if I had not intuited the exact gift they *really wanted for Christmas. What a life saver! And it could be bought at the after-Christmas sale too!
But mostly it is a way to extend the Christmas spirit into the new year. I am always pleased my sons remember that childhood experience when they contact me on Jan. 6th to wish me a "Happy Little Christmas."