One answer would be that it means whatever we want it to mean, but that's a little too easy, for something so powerful, even when it is not acknowledged to be so.
For a declining but still substantial number of us, it is simply the story of the start of the Incarnation, Word made Flesh. The story, to be found in Matthew and Luke, is literally true, for such people. (Mark and John don't mention it.) Angels from the realms of glory, a virgin birth, shepherds and Herod, the whole thing. It happened, just as the Battle of Actium or the Act of Union happened.
At the other end of the scale, there's the entirely commercial and secular view; a chance to buy and sell, to eat and drink stuff we don't usually eat and drink, and to get presents (OK, to give them, too.)
My feeling is that a lot of us are in the middle. It's a tremendously powerful myth, a midwinter (almost) celebration that brings family and friends together. But there's more. There is something strong and special about this - the serenity, the idea of light and hope at the darkest time of the year. "In The Bleak Midwinter..."
And for many of us, these powerful feelings seem to survive horrible journeys in horrible weather, gruesome presents from irritating relatives, and all the other usual seasonal grumbles with which people like to shield themselves from the idea of giving in to a little joy.
But one casualty of the modern Christmas, other than the New Year's Eve drunks being wiped down and sewn up in A&Es across the land, is the idea of the Twelve Days of Christmas, of Christmas as a season, not just a weekend. It vanishes into sales, measurements of how well The High Street did, trade figures and news of dreadful but relatively small-scale events here and elsewhere - events that happen all the time, but they are usually covered over by The News. Well, The News must go on being reported, even when there isn't much News, so: how about an unusually vile murder?
So: hooray for Twelfth Night, for Epiphany and the Kings. Next Christmas, let's try to keep that special feeling running until at least the day after Twelfth Night!
In this spirit, then, Threnody will be learning "The Twelve Days of Christmas" and singing it in parts.