The twelve days begin, as I understand it, on Christmas day and end when the Magi (three kings or three wise men) come into the presence of the infant Jesus (symbolic of Jesus, the Christ, becoming made known to the non-believing world*) on the feast of Epiphany (5th January).
Many people have told me that the words of the 'twelve days' song have hidden meaning (but don't so many of the old nursery rhymes as they deal with catholicism, the plague and the like?) but everything I've found seem to indicate that this is perhaps an extended myth or the 'urban' kind - That said the explanations out there are wild, weird and varied. Here's a compliation of the most consistent view of them:
Partridge in a pear tree Jesus Christ
Two turtle doves Old and New Testaments
Three French Hens Faith, hope, love (1 Cor 13) or the Trinity
Four Calling birds The Four Gospels or four Evangelists
Five Golden Rings The Pentateuch (Genesis to Deuteronomy)
Six geese a laying Six days of creation
Seven Swans a swimming Seven Gifts of the Holy Spirit
Eight maids a-milking Eight Beatitudes
Nine Ladies Dancing Nine Fruits of the Holy Spirit
Ten Lords a-leaping Ten Commandments
Eleven pipers piping Eleven faithful disciples
12 drummers drumming Twelve articles of the Apostles Creed
Like the old man 'who wouldn't say his prayer' the song apparently has origins in the Catholic persecution and is a way of remembering the catechism in a safe manner. Regardless - it's a fun song, isn't it?
And here we are at Day one: All about Jesus - innit?
* The shepherds were Jews and they represent Jesus being revealed to the people of faith.
The Magi, being non-Jews, represent Jesus being revealed to the heathen (meaning non-Jewish)