According to Shammai, one begins with the "days remaining;" that is, with the "maximum potential" of the commandment. According to Hillel, one begins with the "days completed;" that is, with the "realized potential" of the commandment. Thus, on the first night, before the kindling, there remain eight days, a potential of eight lights, so, according to Shammai and his Academy, the correct number of lights is eight. According to Hillel and his Academy, before the kindling, we say that tonight will be the first night of realized potential for this commandment, so we light one.
And so on till the last night, where, according to Shammai, there is only one night remaining, so we light just one light, whereas according to Hillel, this will be the eighth night of realized potential, so we light the full complement of eight.
Each viewpoint has either a model from the Bible or a strong logical argument to buttress their opinion.
Bait Shammai has the model of the offering of bulls during the Festival of Succot, which begins with thirteen on the first day and, decrementing by one each day, finishes with seven on the seventh day (for a total of seventy, corresponding to the "seventy nations of the world," for whose benefit the offering is made.). And decrements yet again to just one bull, on the "eighth day" of Succot, Shmini Atzeret, which corresponds to the singular People of Israel.
Bait Hillel has on its side the general rule followed in many areas of the Torah, that "Ma'alin Ba'Kodesh ve'ayn Moridin," One increases in matters of holiness, and does not diminish.
As mentioned in the "Basics" Section, the matter of the dispute was voted upon, in the democratic spirit of the Talmud, and the Halachah (the practice to be followed in the actual case) was decided in accordance with Bait Hillel; namely, to begin with one and conclude with eight lights on Chanukah.
Also as mentioned in the "Basics" Section, Hillel and Shammai were frequent disputants, differing on fundamental questions of Jewish Law. It is about disputes such as these, where both parties are striving for the "sake of heaven," that Pirkei Avot (a Tractate of the Mishnah, dealing with basic ideas of Judaism) says that both sides of the argument will live forever, because "Elu V'Elu Divrei Elokim Chayim," "Both the one and the other side of the argument are the words of the Living G-d."
How can this be? Does this complementarity suggest that there are two standards of truth?
The answer seems to be along these lines: The Talmud says that even though in most cases, the Halachah is decided in accordance with the view of Hillel, in reality, Shammai's analyses were sharper and deeper. So why then was the Halachah decided in favor of Hillel?
This is because Hillel's viewpoint is appropriate for the Pre-Messianic Era, the period, basically, of Exile and Persecution, when people are distracted from the ultimate realities of life, which are indeed in accord with the thought of Bait Shammai. The Messianic Era will be a time of actualized potential for humanity, a time of "U'Maleah Ha'Aretz Deah es Hashem Ka'Mayim la'Yam Me'chasim," "Knowledge of the L-rd will fill the world, as water covers the bed of the sea."
And paradoxically, it is then, at the time when mankind reaches the point which Bait Hillel had advocated all along was the correct way to view a Commandment, that the Halachah will revert to the opinion of Bait Shammai. For humanity then will be on a higher level, and be able to apprehend reality as Shammai did.