I read that women are unclean for two weeks after giving birth to a girl and seven days after giving birth to a boy. What does it tell us?
I recommend that you take a look at the commentary of R' Samson Raphael Hirsch on Leviticus 12, and in particular 12:5. I'll try to summarize the pertinent points here, but I won't do his beautiful thoughts and words justice.
The "uncleanliess" you're referring to is "tum-a." According to R' Hirsch, tum-a is a mental condition that would prevent a person from participating in holy practices with the correct mindset. In particular, many types of tum-a cause this problem by afflicting a person with the illusion that people are trapped by the physical world and have no genuine free will. One can't participate in the holy service, in which one dedicates some aspect of oneself to God, without a complete sense that one is approaching God out of free choice. The paradigm of this type of tum-a is contact with a human corpse, which can give a person the depressing impression that people have no more value than this inanimate body.
In the process of giving birth, a woman necessarily surrenders to the overwhelming physical process and thus is intimately subject to the illusion that she is an unfree object of natural forces rather than the holy, volitional being that she is. Until enough time has passed for the depressing effects of this spiritual trauma to ebb, she needs to take a break from holy offerings that require a whole spirit. That's the tum-a that she experiences. The first stage of such recovery takes seven days.
That's for the mother. Of course, the child who's born also goes through a traumatic physical process and what's more, needs to be inducted for the first time into the world of free-willed submission to God. For this purpose, we have different processes for a son and for a daughter. For a son, we do circumcision (12:3), which R' Hirsch says represents "the free willed moral subordination of our physical bodily sensuality under the laws of God" (see his commentary on Genesis 17 for more).
For a daughter, the mother goes through the tum-a recovery period for a second time. It will be the mother's job, in addition to serving as a free-willed moral Jewish Woman in her own right, to also serve as a model of that role for her new daughter. "... and at the birth of each fresh daughter has doubly to arm herself, for the child and for herself, to tread the lofty path of purity and morality up to the heights preached by the Sanctuary of God."
One way to look at Tuma is that it is a lack of holiness. When a women gives birth she has less life inside her, and thus less holiness.
When she gives birth to a girl she lost more holiness than with a boy because the girl inside her also has the capacity to grow life.
The gemoro niddah 31b asks this question. The gemoro answers that tuma is dependant on when she will accept her husband back again. Rather a new notion for it. That the idea of tuma is to stop her being available for her husband so that he should not be able to 'force' or 'pressurise' her into it. And for a boy where everyone is pleased it is after seven days and for a girl fourteen.
The status of "tumah" is not meant to imply sinfulness, degradation or inferiority. On the contrary, it emphasizes, in particular, the great level of holiness inherent in woman's G‑dly power to create and nurture a new life within her body, and the great holiness of a husband and wife's union, in general. Since a woman possesses this lofty potential, she, also bears the possibility of its void. http://m.chabad.org/library/article_cdo/aid/510244/jewish/Ritual-purity-after-birth-of-males-and-females.htm