Albert Barnes Commentary Leviticus 12:6-8

"And when the days of her purifying are fulfilled, for a son, or for a daughter, she shall bring a lamb a year old for a burnt-offering, and a young pigeon, or a turtle-dove, for a sin-offering, unto the door of the tent of meeting, unto the priest: and he shall offer it before Jehovah, and make atonement for her; and she shall be cleansed from the fountain of her blood. This is the law for her that beareth, whether a male or a female. And if her means suffice not for a lamb, then she shall take two turtle-doves, or two young pigeons; the one for a burnt-offering, and the other for a sin-offering: and the priest shall make atonement for her, and she shall be clean." — Leviticus 12:6-8 (ASV)

The sacrificial act expressed an acknowledgment of sin and a dedication of herself to Yahweh. See Leviticus 8:14.

(Leviticus 12:6)

Of the first year - literally, as in the margin, “a son of his year.” This expression is supposed to mean one less than a year old, while “the son of a year” is one that has just completed its first year.

(Leviticus 12:8)

A lamb - Rather, one of the flock; either a sheep or a goat; it is not the same word as in Leviticus 12:6.

Two turtles, or two young pigeons - See the note at Leviticus 1:14. The Virgin Mary availed herself of the liberty which the Law allowed to the poor and offered the inferior burnt-offering (Luke 2:24).