I. The Prohibition
The Mishnah (Sotah 20a) quotes R. Eliezer who states that one who teaches his daughter Torah is as if he had taught her tiflus (I'll leave that untranslated and we can just assume that it is a bad thing). The Shulhan Arukh (Yoreh De'ah 246:6) quotes this law and states that this applies only to the Oral Torah but one should still not teach women the Written Torah either. However, the Rema points out that women need to learn the basic laws that they must fulfill and the Taz (ad loc., 4) argues that women are also allowed to learn the simple meaning of the Written Torah.
The conclusion is that there are four areas within this law:
1. Women may not learn the Oral Torah
2. Women may learn the simple meaning of the Written Torah
3. Women may not learn the Written Torah in depth
4. Women must learn the laws that apply to them
The posekim assume that included within "the laws that apply to them" is mussar that keep women them within the bounds of halakhah. Even the Satmar Rav, who as we shall see was very strict on these rules, permits women to learn mussar (Va-Yo'el Moshe, Ma'amar Lashon Ha-Kodesh ch. 33). He does not, however, permit women to study even Rashi on the Torah because it contains Oral Torah.
Note that the suggestion that this prohibition emanates from some sort of misogynist rabbinic bias or historical circumstance is insulting and bordering on heresy.
II. Motivated Women
R. Hayim Yosef David Azulai, the Hida, wrote (Tuv Ayin, no. 4) that all of the above only applies to a woman who does not want to study. We cannot force her to learn, like we do to schoolboys. However, if she wants to learn then not only may she do so on her own, but we may teach her. According to the Hida, the prohibition does not apply to a motivated woman or girl. (Although this does not go undisputed.)
Indeed, in truth there is a prohibition against teaching Torah to any student -- male or female -- who one knows is not properly prepared and motivated, a talmid she-eino hagun (Shulhan Arukh, Yoreh De'ah 246:7). The Gemara (Berakhos 28a) relates that Rabban Gamliel would announce that any student whose is not pure enough that his outsides are like his insides may not enter the study hall. While this approach -- requiring purity -- was rejected, a middle approach was adopted as standard. Namely, if one knows that a particular student is definitely bad then he may not be taught. It seems that for women there is a higher standard and she must be motivated in order to have this permission to learn.
Regardless, in terms of institutionalizing teaching in schools, in which all girls are taught equally, this law must be taken into account. According to the letter of the law, girls in school may only be taught the laws that apply to them, the plain meaning of Scripture, and mussar.
III. Bais Ya'akov
However, the nineteenth and early twentieth centuries saw such upheavals in the Jewish world that it was deemed necessary to institute a uniform and deep education of women so they will appreciate and maintain the Jewish tradition. In one sense, this is a clear violation of the above law that was necessarily violated. In another, however, it is not in that this all falls under the study necessary for women to observe the laws that apply to them. In previous generations, women only needed to be taught -- even informally -- what their religious oblgiations were in order for them to fulfill them. In more recent years, women needed to be taught more about Judaism in general in order for them to fulfill those laws. If that is what is necessary, then it is permissible.
The following are the words of R. Yisrael Meir Kagan, the Hafetz Hayim (Likkutei Halakhos, Sotah p. 21):
It would appear that all this was intended for earlier generations when everyone dwelt in the place of their familial ancestral home and ancestral tradition was very powerful among all to follow the path of their fathers... under such circumstances we could maintain that a woman not study Torah and, for guidance, rely on her righteous parents, but presently, due to our myriad sins, ancestral tradition has become exceptionally weak and it is common that people do not dwell in proximity to the family home, and especially those women who devote themselves to mastering the vernacular, surely it is a great mitzvah to teach them Scripture and the ethical teachings of our sages such as Pirkei Avos, Menoras Ha-Ma'or and the like so that they will internalize our sacred faith because [if we do not do so] they are prone to abandon the path of God and violate all principles of [our] faith.This was a founding statement of the Bais Yaakov movement. Based on this philosophy, and with the backing of other Torah giants, schools were established to give girls a thorough education in Judaism. The amount of Oral Torah taught was not extensive, certainly not in original texts, but it did include commentaries on the Torah such as Rashi. (It is to this phenomenon that the Satmar Rav objected in his above-quoted book [chs. 48-50], even claiming that the Hafetz Hayim was on his side!)
IV. Learning Gemara
Extending this principle, R. Joseph B. Soloveitchik advocated the institutional teaching of Gemara to women so that they gain an appreciation of Judaism and do not, God forbid, leave complete Jewish observance. Here is how his grandson explained this approach (R. Mayer Twersky, "A Glimpse of the Rav" in R. Menachem Genack ed., Rabbi Joseph B. Soloveitchik: Man of Halacha, Man of Faith, p. 113):
The halakha prohibiting Torah study for women is not indiscriminate or all-encompassing. There is complete unanimity that women are obligated to study halakhot pertaining to mitsvot which are incumbent upon them... The prohibition of teaching Torah she-Ba'al Pe to women relates to optional study. If ever circumstances dictate that study of Torah sh-Ba'al Pe is necessary to provide a firm foundation for faith, such study becomes obligatory and obviously lies beyond the pale of any prohibition. Undoubtedly, the Rav's prescription was more far-reaching that that of the Hafets Hayim and others. But the difference in magnitude should not obscure their fundamental agreement...
V. Online Resources
Torah Perspectives on Women's Issues by R. Mayer Twersky
Torah Study for Women by R. Aharon Lichtenstein
A Perspective on Women's Bina Yeteira by R. Walter Wurzburger
Talmud Study by Women by R. Yehuda Henkin
Teaching Torah to Women by R. Moshe Weinberger
The Obligation Of Talmud On Women According To Maimonides by Dr. Warren Zev Harvey
Jewish Education for Women by R. Moshe Kahn
The Frumteens moderator, a big fan of the Satmar Rav, makes the following largely correct but also entirely misleading claim:
The prohibition of learning Gemora is not my belief, but chazal and the shulchan aruch.Here again, the moderator is entirely correct while still being totally wrong. First of all, he neglects to mention that there are authorities who allow women who desire, on their own initiative, to study Talmud to do so. According to these authorities, the only problem is teaching women/girls who do not necessarily want to learn the subject. He is correct that there is a prohibition against women studying Torah, however that only applies to women who are not motivated in themselves. It is not a lie that women who want to may learn Gemara. If he believes that this is so, then I suggest he have a talk with R. Yitzhak Hutner's daughter Rebbetzin Beruriah David, whose father taught her the Talmud much better than the Frumteens moderator knows it.
Depite your claim of many authorities stating that women are allwoed to learn Gemora nowadays, there are none. None that have any authority to argue with a unanimous halachah in Chazal, rishonim and shulchan aruch - yes, unanimous.
They taught you a lie. Simple as that. You will find me zero - not a single opinion that women may be taught in a classroon setting torah shebal peh. Oh, I know that cnotemporary Modern Orthodox rabbis have said so, but they themselves are the ones that need to find the authorities that permit, and not a single one of them had.
However, he then switches to institutional study. Here, he is correct that there are no authorities in the past who permit teaching Talmud to women in a classroom setting. No one is disagreeing with that, just like the Hafetz Hayim would not disagree that his innovation had no previous sources to defend it. Both are innovations and both are in the same spirit. No one is claiming to the contrary. One can disagree, like the Satmar Rav, with the theory behind the Bais Ya'akov movement just like one can disagree with R. Soloveitchik's theory about teaching Talmud to women. But the statement that there are no earlier sources that permit the matter is entirely correct and entirely irrelevant.