Orthodox Judaism has a reputation for being very patriarchal and unequal towards women. I certainly feel like a lesser member of society when banished behind the mechitsa (barrier to separate men and women), not allowed to sing in synagogue (because a woman's voice might tempt the men), and not allowed to become a rabbi.
Not everyone feels that traditional Judaism is misogynistic--those people primarily being the Orthodoxy themselves. Women are regarded as being inherently more spiritual beings than men. Following that reasoning, women don't NEED to read from the Torah in the synagogue, be counted in a minyan (10 men must be present for prayers to be held), or take part in other aspects of public religious life. Unlike much of the rest of the world, in Judaism women were never the property of their husbands. A woman can never be forced to have sex against her will, even within marriage. And women have always had the right to work outside the home (sometimes this is a necessity, as when their husbands are studying Torah all day).
The battle that is being waged has very clear boundaries, although they are extremely complicated for an outsider to decipher. First, one has to understand the nature of Jewish law. There is the written law (the Torah and other holy writings.) Second, there is the oral law which was supposedly also given to Moses on Sinai and then passed down through the years by various rabbis. It was codified into the Talmud. Basically, the written law is the big cheese not to be messed with at all, and the oral law is a fluid set of laws that can be argued about. Legal precedent is a big deal (ever wonder why there are so many Jewish lawyers?).
Some things are outright prohibited (eating pork or lighting a fire on Shabbat, for example). Some things are required (men must produce children). Some things are not required but encouraged (women having children). Some things are not forbidden by law but not outright permitted (MTV or lesbian sex). Some things are required for men but optional for women (wearing a prayer shawl). For Orthodox feminists, they are interested in the things that are not outright forbidden but, due to cultural reasons, are no longer done by women. They comb the texts for examples of Jewish women giving sermons, being kosher butchers, or wearing prayer shawls.
So why can’t we just observe at the level we feel most comfortable? In Jewish law, when something is optional but a critical mass of people feel it is necessary, it becomes obligatory. For example, there is no law saying that Jewish men should wear a kipa (yarmakule or skullcap), but you won't find a religious Jewish man without one. In the same sense, women who want to "be better Jews" by following more of the Jewish law run up against opposition by other Orthodox women. These women have a lot of children, most are the sole breadwinners in the family, and they frankly don't have the time or energy for additional obligations. The idea of "live and let live," letting the women who want additional responsibility have it and letting others stay the way they are, is an idea that simply does not work for the Orthodoxy. If the feminists believe in live and let live, it weakens their position. If a woman truly believes it is a powerful blessing to perform additional duties, then logically she should want all women to do it.
In Israel, Orthodox feminism has focused on two spheres: religious education and legal issues. In America, Jewish feminism has focused on the public sphere, specifically prayer services. Even if one works within the framework of "permitted but not done," such as reading from the Torah in public prayers or giving a sermon, they have met with fierce opposition. Many women are extremely frustrated and find they cannot both be part of the Orthodox movement and the feminist movement.
Partially in response to the American approach, Israeli Orthodox feminists have focused on the private sphere. There is the belief that anything fought for in the public sphere has some element of egoism and self-interest, whereas things in the private sphere have a purer intention. These women have argued for studying the Talmud (Jewish oral law). Orthodox women almost never receive a formal education in Jewish law--even though it is their responsibility in the family to pass on Judaism to the next generation. Many Orthodox men object, but others don't see the harm in women learning the rules of their own religion. Many have become Talmud educators, all within the Orthodox sphere--and it has been a big success. Fighting to learn Jewish law has had massive advantages when dealing with the legal sphere.
Even though parts of Israel are modern, part of the developed world, and players on the world sphere, much of Israeli family law is governed by the ultra-Orthodox, grounded in documents that were written thousands of years ago. No other society I know of--not British, not Chinese, no one--relies on laws that are that old. Regardless if you are religious or secular, in Israel you obey the Orthodoxy when it comes to family law. This includes weddings, divorces, inheritance, and other similar matters. So you can parade around in assless chaps all over Tel Aviv or never set foot in a synagogue in your whole life, but on the day of your wedding it will be presided over by an Orthodox rabbi.
First, a success story: under Jewish law, sons and daughters should not inherit equally. However, the government of Israel takes the position that progeny should inherit equally. Because this is a Jewish country, one cannot simply disregard a Jewish law. So in Israel, wills can be written that "gift" equal portions of an estate to their children. It's similar to how some Americans try to get out of paying estate taxes by deeding or gifting things to their kids.
Now, one of the big problems: under Jewish law, a woman cannot get a divorce unless her husband grants it. However, you cannot force a man to get a divorce or it is invalid. This has been a big problem for years--men disappear for whatever reason (accidents, MIA in the army) or outright refuse to grant a divorce, and there's nothing the woman or the state of Israel can do to grant that woman the right to remarry. The husbands who outright refuse present a particularly frustrating problem. Maimonides, one of the most famous Jewish scholars, advocated beating the crap out of a man until he agreed to a divorce. His argument is that a man isn't just holding a woman hostage, not letting her remarry or move on with her life, but he is holding the entire community hostage.
In Israel, they don't beat the crap out of people, but they can take away a man's money and chuck him in prison until he grants a divorce. Even at that point, some men still refuse, so there is a small population of women in Israel who are really at a loss as to what to do. The feminists took a very creative approach to this dilemma--they can't change Jewish law, but they can advocate that women ask for a pre-marital agreement in which the power of granting a divorce is turned over to the Jewish courts. Again, because they studied their Jewish laws, these women have precedents. Apparently, King David ordered all of his soldiers to write declarations that if they didn't return home within a certain period of time, their wives are automatically granted a divorce. And if it's good enough for King David, it's good enough for Joe Shlomo.