1) All Jews are obligated in the Mitzvah of lighting Chanukah candles.
This applies to men, women and children (who have reached the age of Chinuch; approximately 5 or 6 years of age), as all Jews were saved in the miraculous victory of the Macabbees over the mighty Greek army. (Additionally, as we discussed in the previous Halachos, women are specifically obligated in the Mitzvos of Chanukah, due to their being saved from the harsh decrees levied against them, via the heroic act of Yehudis, a woman)
2) The prevalent Minhag is for the male head of household to exempt all the female members of the house (adults and children alike), through the lighting of his menorah.
The females being exempted should make sure to be present and listen to the recital of the Brachos [and answer Amen] while the menorah is kindled. (Mishna Berurah 675:9)
Of course, if no man is available to exempt her, a women may light [and make the blessings over] her own menorah. A woman can also exempt other people with her lighting, when no man is available to do this. (See Smag Hilchos Chanukah, Taz 675:3, Magen Avraham ibid.)
Boys who have reached the age of Chinuch should preferably light their own menorahs [with all the Halachic details involved] and not rely on the head of the household’s lighting. (Ram’a 675:3. See also Mishna Berura and Biur Halacha ibid. Dibur Hamaschil U’Ldidan, regarding if there is a Mitzvah of Chinuch for “Hidur Mitzvah” or just for the “letter of the law” and thus children would only have to light one candle per night.)
(Halachos for Today)