Although lighting the Menora is a time-bound mitzva which women are normally exempt from, the Gemara (Shabbos 23a) writes that as women were also involved in the miracle of Chanuka, they are obligated in ner Chanuka. Thus, women must light the Menora unless someone else has lit one at home.
The Rishonim debate what miracle the Gemara is referring to. According to Rashi this refers to the abolishment of the threat that brides were under while according to Ran it was the famous incident of Yochanan’s daughter, Yehudis, who killed the Chief General. As she fed him cheese to make him thirsty, the Rema (OC 670:2) writes that there is a minhag to eat milky foods on Chanuka. R’ Yaakov Emden (Mor Uketzia OC 670) points out, however, that the story of Yehudis occurred hundreds of years earlier during the time of the first beis hamikdash (See Kaf Hachaim OC 670:17; Aruch Hashulchan OC 670:8; Rivevos Ephraim 4:157).
Additionally, the Shulchan Aruch (OC 670:1) writes that women should not do any melacha while the Menora lights are burning. The Taz (OC 670:2) compares Yehudis’ actions to the women who didn’t participate in the golden calf, for which they were rewarded with Rosh Chodesh. Accordingly, this restriction only applies to women.
The Vilna Gaon (Biur Hagra OC 670:1) writes that this is to remind one not to benefit from the Menora’s lights. R’ Ephraim Greenblatt (Rivevos Ephraim 5:434 beshem his father) writes that accordingly, this restriction applies equally to men. The minhag however, is that only women need to refrain from melacha (See Mor Uketzia ibid).
While the Magen Avraham (OC 670:2) writes that this applies so long as the lights are burning, the Mishna Berura (670:4) and Kaf Hachaim (OC 670:8) write that this only applies for the first half hour.
R’ Ephraim Greenblatt (Rivevos Ephraim 1:436; 3:163) writes that while there are different opinions as to what melachos are permitted, only sewing and laundry, etc. are forbidden. One may cook and do any other melacha that is otherwise permitted on chol hamoed.R’ Shlomo Zalman Auerbach (Minchas Shlomo Tinyana 58:5) writes that women do not need to recite hallel on Chanuka while R’ Shmuel Wosner (Shevet Halevi 1:205) writes that women are obligated.