Adam Clarke on Leviticus 12:6

When the days of her purifying - It is not easy to account for the difference in the times of purification, after the birth of a male and female child. After the birth of a boy the mother was considered unclean for forty days; after the birth of a girl, four-score days. There is probably no physical reason for this difference, and it is difficult to assign a political one. Some of the ancient physicians assert that a woman is in the order of nature much longer in completely recovering after the birth of a female than after the birth of a male child. This assertion is not justified either by observation or matter of fact. Others think that the difference in the time of purification after the birth of a male and female is intended to mark the inferiority of the female sex. This is a miserable reason, and pitifully supported.

She shall bring - a burnt-offering, and - a sin-offering - It is likely that all these ordinances were intended to show man's natural impurity and original defilement by sin, and the necessity of an atonement to cleanse the soul from unrighteousness.