The Shulhan Aruch (Orah Haim 303:26; listen to audio recording for precise citation) rules that it is forbidden for a woman to braid her hair, or to undo a braid, on Shabbat. The Mishna Berura (commentary by Rav Yisrael Meir Kagan, 1839-1933) explains (listen to audio recording for precise citation) that braiding hair is forbidden on Shabbat by force of Rabbinic enactment, and not on the level of Torah law. The Torah prohibition of Oreg (weaving) applies only to weaving material that is not attached to its source. Weaving, of course, is done only after the wool has been removed from the animal, and therefore the Torah prohibition of Oreg does not apply to braiding hair attached to one’s head. Furthermore, a hair braid is only temporary, and for this reason, too, it does not entail a Torah prohibition on Shabbat.

The reason why the Sages forbade braiding hair on Shabbat, the Mishna Berura explains, is because braiding hair resembles building. Indeed, the Torah (Bereshit 2:22) relates that God "built" the rib taken from Adam into a woman, and Hazal interpreted this to mean that He braided Hava’s hair before bringing her to Adam. The Sages further noted that in certain regions, the same word is used for "building" and "braiding" ("Binita"). In light of this association between braiding and building, the Sages enacted a prohibition against braiding hair on Shabbat, when building is forbidden.

The Mishna Berura adds that braiding the hair of a wig – or undoing a braid on a wig – is also forbidden on Shabbat. Even though this would not violate the Torah prohibition of Oreg, since the braid is not permanent, nevertheless, it is forbidden by force of Rabbinic enactment. The Mishna Berura notes that one should not object to women braiding wigs if he anticipates that they would pay no heed to his admonition. In such a situation, we follow the rule that it is preferable to allow somebody to commit a violation unintentionally than put him in a situation where he would do so intentionally.

Clearly, though, those who seek to properly observe the Halacha should be instructed to make or undo their braids, on their hair or their wigs, before Shabbat.

Summary: It is forbidden on Shabbat to make or undo a braid on a woman’s hair or a wig.