Women are not obligated to wear a tallit. This is because they are exempt from fulfilling almost all time-bound positive commandments (such as reciting the Shema, which is done morning and night, or taking the Four Kinds on the holiday of Sukkot). Nonetheless, women do fulfill many of these mitzvot if they so desire.
Yet the prevailing custom is that women do not wear tallitot. A number of reasons for this reticence are found in Halachic works:
A. Both women and men are Biblically forbidden to wear clothing normally associated with the other gender. For example, men may not wear skirts. Since a tallit is traditionally a male garment, for a woman to wear one would constitute a violation of this statute.
B. Although women observe many time-bound mitzvot though they are not obligated to do so – an admirable practice for which they are certainly greatly rewarded – a tallit is different because there is no obligation whatsoever to wear a tallit—even for a man. Rather, in the event that he wears a four-cornered garment, a man must attach fringes to its corners. Since a man is not obligated to seek out such a garment, women who are entirely exempt from this mitzvah (i.e. they may wear fringeless four-cornered garments) do not wear them at all.
C. A woman who fulfills this mitzvah, which she is not obligated in doing and is not performed by the vast majority of her gender, draws undue attention to her excessive piety in an inappropriately ostentatious manner. [The concept of abstaining from a particular activity because it is deemed to be ostentatious is a general rule in Jewish law, applied both to men and women in various cases.]
D. On a mystical level, the inner workings of this mitzvah are male oriented and just don't "do it" for a woman.
So what is a woman who wishes to wear a tallit to do?
Rabbi Moshe Feinstein, eminent 20th century halachic authority, writes that a woman who desires to wear a tallit may do so, provided that she wears a distinctively feminine tallit, to avoid the problem mentioned above. He cautions, however, that this applies only to women whose desire to wear a tallit stems from a yearning to fulfill this mitzvah, though recognizing that they are not required to do so, and not to individuals who don a tallit as a "protest," a means of challenging what they perceive to be a gender bias in Jewish law. Such an individual is not fulfilling a mitzvah, and to the contrary.
While the above addresses the practical aspect of this question, I would be remiss if I did not address the deeper issue this question involves. While altogether the feminist movement is to be commended for the equal rights it has secured for women, and the elevation of the woman's social, legal and economic status, a certain aspect of this movement's aims is questionable at best. I refer to the desire to make women masculine, rather than accentuate their feminine qualities. To evaluate a woman based on her ability to "do whatever a man can," is to dishonor womanhood, and all the unique qualities it brings to the table. A true feminist is someone who believes and is committed to making others understand the equality and importance of a women and the natural feminine role, not someone who believes that women should forsake their femininity in favor of becoming more man-like.
The same is true in the religious arena. There is a certain element that wishes to see equality between man and woman in all areas of religious ritual—i.e. that women should do whatever men do. The apparent premise of this movement is the belief that the woman's role in Judaism is less important and noble than the man's, and thus the need to right this perceived wrong.
But the One who created both man and woman thinks otherwise. He is aware that He endowed man and woman with equally valuable but fundamentally different qualities and talents—and then in His Torah advised both man and woman how to maximize these unique strengths.
So the larger question is: why would a woman want to wear a tallit if the Torah does not encourage her to do so?
For more on this topic, see Women in the Synagogue, or browse the articles in our Women, Femininity & Feminism section.