Leviticus 5: Two Turtle Doves

“But if one’s means do not suffice for a sheep, that person shall bring to God as the penalty for that of which one is guilty, two turtledoves or two pigeons—one for a sin offering and the other for a burnt offering” (verse 7).
וְאִם־לֹא תַשִּׂיג יָדוֹ דֵּי שֶׂה וְהֵבִיא אֶת־אֲשָׁמוֹ אֲשֶׁר חָטָא שְׁתֵּי תֹרִים אוֹ שְׁנֵי בְנֵי יוֹנָה לַה׳ אֶחָד לְחַטָּאת וְאֶחָד לְעֹלָה׃
The ability to bring a sin offering is an essential part of asking for forgiveness. The proper animal to bring is a sheep, but God does not want anyone to feel priced out of the sin offering. Therefore, someone who cannot afford a sheep is allowed to bring two turtledoves. Someone who cannot afford even the two doves is instructed to bring a tenth of an ephah of flour.
Working backwards—why does someone bring a tenth of an ephah? Ibn Ezra explains that this was the measurement of one day’s worth of flour. Why two birds? Ibn Ezra explains here that one bird is for the offering itself and the second bird is for the fats that were burned on the altar. This way the two birds make up for the two korban functions derived from the one sheep.
The Sifra explains that if a person only had a sheep at home, but if they used it for the korban they would have to borrow money for daily expenses, then they can bring the doves instead. Rabbi Soloveitchik explains that the two birds are linked. Even though one actually is sacrificed as a burnt offering and one as the sin offering, they are considered a pair. Therefore, if someone pledged the first bird as a burnt offering and then died before designating the second bird as a sin offering, the heirs must still bring the second bird. Normally, we do not require that heirs bring a sin offering for a deceased person as the offering could no longer atone for sins. In this case, however, the two birds are considered one single offering split into two types.
As a side note, there is no inherit connection between the two turtle doves here and the Christmas song- twelve days of Christmas. Popular myth has the song as a secret Catholic catechism in protestant England with each item a reminder of Catholic teachings- the two turtle doves as the old and new testament. But there is no real historical evidence for this, and it’s most likely the song was a simple memory game played with children.