The holiday of Hanukah begins on the 25th of Kislev and continues for eight days. As the Gemara in Masechet Shabbat (21) relates, on the 25th of Kislev the Hashmonaim defeated the Greek oppressors and rededicated the Bet Ha’mikdash. As part of this process they kindled the Menora with the only jug of pure oil they found, and the candles of the Menora miraculously burned for eight days.
The Bet Yosef (commentary to the Tur by Maran, author of the Shulhan Aruch) raised the question (in Orah Haim, 670) of why we celebrate this miracle for eight days. The Gemara clearly states that the jug contained enough oil to sustain the candles for one day. If so, then the first day’s burning was not miraculous at all, and the miracle was in fact only seven days. Why, then, do we celebrate Hanukah for eight days?
The Bet Yosef suggests three answers. First, he writes that the Kohanim anticipated that it would take eight days for new pure oil to arrive, so already on the first day they divided the small jug of oil into eight parts, and lit the candles with one part each day. The miracle was thus that each day, one-eighth of the jug of oil sustained the candles for an entire day, a period that normally required an entire jug. Hence, the miracle in fact spanned the entire eight-day period.
Secondly, the Bet Yosef suggests that when the Kohanim entered the Mikdash each morning, they noticed that the jug remained full – even though they had used all its oil for kindling the Menora the night before. This occurred each morning for eight days, and thus the miracle was indeed an eight-day event. Finally, the Bet Yosef writes that perhaps the lamps of the Menora remained filled with oil each morning, and the miracle was thus that the oil in the lamps burned without being depleted. This, too, would account for the eight-day celebration.
The holiday of Hanukah was enacted by the Sages, and thus does not have the status of a Torah obligation. For this reason, we do not add a ninth day to this holiday in the Diaspora, the way we add a day to other holidays. This extra day is observed only in conjunction with holidays established by the Torah, as opposed to the Rabbinically-ordained festivals of Hanukah and Purim.
The Shulhan Aruch (Orah Haim 670:1; listen to audio recording for precise citation) writes that it is forbidden to fast or eulogize on Hanukah. Working, he writes, is permissible, though there is a custom for women to desist from working while the Hanukah candles burn (or at least during the first half-hour after they are lit). The Magen Abraham (Rabbi Abraham Gombiner, Poland, 1637-1683) explains that this custom commemorates the story of Yehudit, a Jewish woman who heroically killed a Greek general. (Some sources indicate that the story of Yehudit did not occur during the Jews’ battle against the Greeks, but rather earlier, during the period of Greek oppression. Either way, her heroism is certainly worthy of commemoration on Hanukah.)
By contrast, the Maharil (Rabbi Yaakov Halevi Moelin, Germany, 1355-1427) held that both men and women should refrain from work while the candles burn. The Bet Yosef explained that this practice serves to demonstrate that the candles were lit to publicize the miracle, and not for personal use. By abstaining from work while the candles burn, we show that we did not kindle them to provide light for our normal activities, but rather to commemorate the great miracle of Hanukah. Additionally, this custom helps ensure that a person spends some time reflecting on the Hanukah miracle. It is thus proper after lighting the Hanukah candles not to immediately return to one’s normal affairs, but to instead sit and spend time celebrating the miracle.
The Kaf Ha’haim (Rabbi Yaakov Moshe Sofer, Baghdad-Israel, 1870-1939) comments (670:10) that there are women who vow during times of danger to refrain from working while the Hanukah candles burn. Many women, he writes, have earned salvation by taking such a vow. It is thus a valuable and worthwhile custom to refrain from work while the Hanukah candles are lit, at least for some of the days of Hanukah.
The Shulhan Aruch rules that there is no obligation to eat festive meals on Hanukah, since this holiday celebrates a spiritual, rather than physical, victory. The Greeks sought not to destroy the Jewish people, but rather to destroy the Jewish religion. This is as opposed to the Purim story, when Haman sought to annihilate us as a people. The physical salvation of Purim is thus celebrated through festive eating and drinking, while Hanukah is observed as a mainly spiritual victory, rather than a physical triumph. Nevertheless, some authorities write that it is admirable to have festive celebrations on Hanukah, to commemorate the completion of the Mishkan’s construction in the wilderness, which took place on Hanukah. The Rama (Rabbi Moshe Isserles, Poland, 1525-1572) writes that if these meals contain words of Torah and songs of praise to God, then they certainly have the status of a "Misva meal."
Of course, on Rosh Hodesh Tebet (which falls on Hanukah) one should have a special meal, just as on every Rosh Hodesh. This obviously applies to Shabbat Hanukah, as well. The Ben Ish Hai (Rabbi Yosef Haim of Baghdad, 1833-1909) records a custom to light a special candle on Rosh Hodesh Tebet in memory of the famous Tanna, Rabbi Meir Ba’al Ha’nes.
There is a custom to eat cheese and other dairy products on Hanukah, since Yehudit (in the incident mentioned above) fed the general dairy products which eventually led to him going to sleep, whereupon she succeeded in assassinating him.