The rabbis ask: What is Hanukkah, and why are lights kindled on Hanukkah? The Sages taught: On the twenty-fifth of Kislev, the days of Hanukkah are eight. One may not eulogize on them and one may not fast on them. What is the reason? When the Greeks entered the Sanctuary they defiled all the oils that were in the Sanctuary by touching them. And when the Maccabees overcame them and emerged victorious over them, they searched and found only one cruse of oil that was placed with the seal of the High Priest, undisturbed by the Greeks. And there was sufficient oil there to light the menorah for only one day. A miracle occurred and they lit the candelabrum from it eight days. The next year the Sages instituted those days and made them holidays with recitation of hallel and special thanksgiving in prayer and blessings.
Al Hanisim (from the Siddur)
"We thank You also for the miraculous deeds and for the redemption and for the mighty deeds and the saving acts wrought by You, as well as for the wars which You waged for our ancestors in ancient days at this season. In the days of the Hasmonean Mattathias, son of Johanan the high priest, and his sons, when the iniquitous Greco-Syrian kingdom rose up against Your people Israel, to make them forget Your Torah and to turn them away from the ordinances of Your will, then You in your abundant mercy rose up for them in the time of their trouble, pled their cause, executed judgment, avenged their wrong, and delivered the strong into the hands of the weak, the many into the hands of few, the impure into the hands of the pure, the wicked into the hands of the righteous, and insolent ones into the hands of those occupied with Your Torah. Both unto Yourself did you make a great and holy name in Thy world, and unto Your people did You achieve a great deliverance and redemption. Whereupon your children entered the sanctuary of Your house, cleansed Your temple, purified Your sanctuary, kindled lights in Your holy courts, and appointed these eight days of Hanukkah in order to give thanks and praises unto Your holy name."
Rabbi David Hartman, Trusting in a New Beginning in A Different Light
In considering the miracle of the cruse of oil, our Rabbis asked why the holiday of Hanukkah was celebrated for eight days rather than for seven days. Since there was, by all accounts, sufficient oil for one day, only seven of the eight days of burning may be designated as miraculous days. Though several ingenious explanations were offered, what strikes me as being the miraculous feature of the initial day was the community's willingness to light the lamp in spite of the fact that its anticipated period of burning was short-lived. The miracle of the first day was expressed in the community's willingness to light a small cruse of oil without reasonable assurance that their efforts would be sufficient to complete the rededication of the Temple. Hanukkah celebrates the miracle expressed by those who lit the lamp and not only the miracle of the lamp's continued burning for eight days.
The Sages taught in a baraita: The basic mitzva of Hanukkah is each day to have a light kindled by a person, the head of the household, for themself and their household.
And the mehadrin, (those who are super strict in the performance of mitzvot), kindle a light for each and every one in the household.
And the mehadrin min hamehadrin, who are even more meticulous, adjust the number of lights daily. Beit Shammai and Beit Hillel disagree as to the nature of that adjustment.
Beit Shammai says: On the first day one kindles eight lights and, from there on, gradually decreases the number of lights until, on the last day of Hanukkah, they kindle one light.
And Beit Hillel says: On the first day one kindles one light, and from there on, gradually increases the number of lights until, on the last day, they kindle eight lights.
Ulla said: There were two amora’im in the West, (Eretz Yisrael). who disagreed with regard to this dispute, Rabbi Yosei bar Avin and Rabbi Yosei bar Zevida.
One said that the reason for Beit Shammai’s opinion is that the number of lights corresponds to the incoming days, i.e., the future. On the first day, eight days remain in Hanukkah, one kindles eight lights, and on the second day seven days remain, one kindles seven, etc.
The reason for Beit Hillel’s opinion is that the number of lights corresponds to the outgoing days. Each day, the number of lights corresponds to the number of the days of Hanukkah that were already observed.
And one said that the reason for Beit Shammai’s opinion is that the number of lights corresponds to the bulls of the festival of Sukkot: Thirteen were sacrificed on the first day and each succeeding day one fewer was sacrificed (Numbers 29:12–31).
The reason for Beit Hillel’s opinion is that the number of lights is based on the principle:
One elevates to a higher level in matters of sanctity and one does not downgrade. Therefore, if the objective is to have the number of lights correspond to the number of days, there is no alternative to increasing their number with the passing of each day.