SONG: The 12 Days of Christmas
While a seemingly silly song - imagine receiving all these gifts from a loved one! - according to legend, this song was a “secret” catechismal tool of the Roman Catholic Church, teaching their members about the faith during a time when the Church was being persecuted in England (by the Protestant Church) from the 16th to 19th centuries. The symbolism of the carol is still useful today to teach Orthodox children (and adults) about the four "calling birds" of the Evangelists, and the three "golden rings" of the Holy Trinity.
You can read more about the significance of the verses here, although note that not all points are necessarily valid to Orthodox Christians.
The Real 12 Days of Christmas
When speaking of the song, it is also worth remembering that these days are not the period before Christmas but rather after. In the Christian Church (both in the Orthodox and Roman Catholic), there is a festal period following the feast day marked by fast-less days/weeks, singing of hymns and songs specific to the feast day (think of how long коляди are sung after Різдво and also купальські after St. John the Baptist (Jul 7) and петрівки after Sts. Peter and Paul (Jul 12), family parties and get-togethers (think carolling etc.), and dances (think Malanka and Гаївки) etc. (the Church also often marks the midway of the festal period and the end, or leave-taking, often with extra services). In the case of the Nativity, these 12 days start January 7th (Dec 25 new calendar) and go until January 18th (Jan 6 new calendar), the 18th being the eve of the feast of Theophany - a strict fasting day.
Our Church, in its usual wisdom, has implemented its own version of the 12 days with different individuals specifically remembered after the Nativity, such as Mary, the Mother of God, the following day (Jan 8). On the following day after a feast day, the Church celebrates the memory of the individual who was (and is) important to the event, and call it the “Synaxis.” In the case of the Nativity, that would be the Theotokos - without whom, we wouldn’t have the birth of Jesus. The “third” day, Righteous Joseph the Betrothed (the Orthodox do not name him as saint, although the Roman Catholics do) is remembered along with St. Stephen, the Protomartyr (that is, the first martyr). The list continues and can be read here on the Greek website.
Sources:
https://www.goarch.org/-/what-are-the-12-days-of-christmas
http://www.scborromeo.org/papers/12days.pdf