Which of the three daily prayers, if any, are women required to recite?
The Gemara briefly discusses this question in Masechet Berachot (20). The Mishna establishes that women are included in the obligation of the prayer, and the Gemara explains the reasoning behind this Halacha. However, there are different versions of the text of the Gemara, yielding very different Halachic conclusions.
According to the version accepted by Rashi (and Tosafot), the Gemara explains that women are included in the obligation of prayer because "Rahameh Ninhu" – prayer is recited not simply to fulfill a technical requirement, but to arouse Hashem’s compassion. Women require Hashem’s compassion no less than man do, and so they are required to pray. It would thus appear that according to Rashi, women’s obligation to pray is no different from a man’s obligation. We would then conclude that they must recite at least Shaharit and Minha each day. Arbit, unlike the other two daily prayers, is, strictly speaking, optional, though men have taken on the recitation of this prayer as an outright Halachic obligation. Women, even according to Rashi, did not take on this requirement, and thus they are not required to recite Arbit.
The Rambam (Rav Moshe Maimonides, Spain-Egypt, 1135-1204) had a different text of the Gemara. He writes in Hilchot Tefila that women are obligated to pray because they are included in every Biblical obligation which is not confined to a particular time ("Misvat Aseh She’lo Ha’zeman Gerama"). The Rambam here follows the view he articulates elsewhere that Tefila constitutes a Torah obligation, but the requirement to recite specific prayer texts at certain times was instituted by the Sages. The Torah obligation of prayer requires one to recite some prayer and some point during the day; it was the Ansheh Kenesset Ha’gedola (Men of the Great Assembly, a special group of Rabbis and prophets established at the beginning of the Second Commonwealth) that instituted the specific requirements of Shaharit, Minha and Arbit. Women are included in the Biblical obligation, which is not dependent on any specific time-frame, just as they are included (as a general rule) in all affirmative commands which have no set time-frame. But they are not included, according to the Rambam, in the requirement to recite the three daily prayers.
Maran (author of the Shulhan Aruch), in his Bet Yosef, brings both versions of the text of the Gemara, but in the Shulhan Aruch (Orah Haim 106:1), he cites the Rambam’s version, that women are obligated to pray because they are included in Biblical commands which are not limited to a particular time. It emerges, then, that according to the Shulhan Aruch, women are obligated to pray only once a day. The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933), expressing Rashi’s view, writes that it is proper for women to pray at least Shaharit and Minha, noting the factor of "Rahameh Ninhu."
The Magen Abraham (Rav Abraham Gombiner, 1633-1683) notes that seemingly, according to the Rambam’s rationale, women are not even obligated to recite a single Amida prayer. After all, as we’ve discussed, the Rambam maintains that women are included only in the Biblical obligation of prayer – which does not require any particular text. The Biblical requirement is satisfied even with the recitation of a brief prayer to G-d when one wakes up in the morning. Strictly speaking, then, according to the Shulhan Aruch’s ruling, it suffices for women to recite even just a brief prayer each day.
Hacham Ovadia Yosef, however, writes that even according to the Rambam’s view, which the Shulhan Aruch accepts, women should recite at least one Amida prayer each day. Firstly, the text of the Amida is exceedingly sacred, laden with secrets and allusions to deep concepts. Secondly, the Rambam does indicate that even to fulfill the Biblical obligation of prayer, one should follow the format of Shebah (praise), Bakasha (requests) and Hoda’a (giving thanks), the structure in which the Amida prayer is arranged. Therefore, Hacham Ovadia rules that women should pray at least one Amida a day, and preferably, this should be Shaharit.
Hacham Ovadia argues in this regard with Hacham Bension Abba Shaul (Jerusalem, 1924-1998), who, surprisingly, writes (in Or Le’sion, vol. 2) that women should recite all three daily prayers, because "Rahameh Ninhu." As Hacham Ovadia notes, this conclusion follows Rashi’s opinion, and not the opinion of the Rambam which is explicitly accepted by the Shulhan Aruch.
There is a separate question as to whether women may or should recite Musaf on Shabbat, Rosh Hodesh and holidays. It appears, at first glance, that the Musaf prayer is purely commemorative, recalling the special sacrifices that were offered on these occasions. Accordingly, Hacham Ovadia ruled that women should not recite Musaf, because they are not obligated to commemorate the Musaf sacrifices. Hacham Ovadia makes an exception with regard to Rosh Hashanah, because Musaf on Rosh Hashanah also includes a petition for Hashem’s compassion, and thus it should be recited by both men and women. Hacham David Yosef, however, in Halacha Berura, writes that women may recite Musaf on any occasion if they so wish. He contends that necessarily, the Musaf prayer also includes an element of "Rahameh" – asking for compassion, because otherwise, it would be recited only by men age twenty and above, as only they paid the annual Mahasit Ha’shekel tax that funded the public sacrifices, and thus only they had a share in the Musaf offering. The fact that even men younger than twenty are required to recite Musaf proves that Musaf is not only commemorative of the Musaf sacrifices in the Bet Ha’mikdash, but also involve "Rahameh," such that women may recite this prayer, too.
The Poskim also addressed the question of whether a woman who has the custom to recite Shaharit and Minha every day should recite an extra "Tashlumin" ("makeup") prayer if she missed one of her prayers. The Mishna Berura, in discussing the laws of the Shabbat candles (263), writes that if a woman is running late on Friday afternoon, and does not have time to both recite Minha and light the Shabbat candles, she should light the candles and then recite an extra Amida at Arbit to make up her missed prayer. Hacham Ovadia disputes this ruling, but what is important for our purposes is that the Mishna Berura clearly recognized the concept of "Tashlumin" even for women. Accordingly, Hacham Bension writes that if a woman on any day missed Minha, she recites an extra Amida at Arbit, emphasizing that this applies even if the woman does not normally recite Arbit. Since she needs to make up her missed Tefila, and the makeup prayer can be recited only in conjunction with the next scheduled prayer, she may recite Arbit so that she can then recite an additional Amida to make up her missed prayer. (This rationale, however, seems difficult, in light of the fact that a "Tashlumin" prayer needs to be recited in conjunction with an obligatory prayer, not a voluntary prayer. We might thus question how a woman can voluntarily recite Arbit in order to then be able to make up her missed Minha prayer.) Interestingly, Rav Yosef Shalom Elyashiv (Jerusalem, 1910-2012) raised the possibility – though he ultimately rejects it – that the woman in this case would recite an additional prayer the next morning, at Shaharit. Since Shaharit is the next required prayer for her, it is then when she makes up her missed prayer.
In any event, Hacham David concludes that if a woman accepted upon herself to recite two or three Tefilot each day, and she happens to miss one prayer, she recites a "Tashlumin" prayer at the next scheduled prayer, such as an extra Amida at Arbit if she misses Minha.
Summary: According to Sephardic custom, women are required to recite one Amida prayer each day, preferably Shaharit. She may, if she wishes, also recite Minha and Arbit. Women should recite Musaf on Rosh Hashanah; there is some debate as to whether they should recite Musaf on other occasions. A woman who has taken it upon herself to recite two or three Tefilot each day, and happens to miss a prayer, recites a "Tashlumin" prayer at the next scheduled prayer. For example, if she misses Minha, she recites two Amida prayers at Arbit, even she does not normally recite Arbit.
