Why are the eight candles on the menorah important for Hanukkah?

The candles of Hanukkah are important because they commemorate the miracle of the oil (Talmud, Shabbat 21b). As an augmentation to the celebration of that miracle, we also thank God during Hanukkah for the Hasmoneans' military victory (during which they ousted the Seleucids and their lackeys, the Hellenizing Jews, making possible the rededication of the Temple).

The Seleucids (Syrian-Greeks) under Antiochus Epiphanes (2nd century BCE), at the instigation of the Hellenizers, had forbidden various Torah-practices such as Sabbath-observance, and pressed the Jews to offer up idolatrous sacrifices. The Hasmoneans (a family of religious Jews) fought to retake the Holy Temple, which had been seized by the Seleucids, and to enable the people to once again observe the Torah. When they reached the Temple grounds, they found only one day's supply of unsullied olive oil, but the oil lamps miraculously burned for eight days (Talmud, Shabbat 21b), allowing enough time for new oil to be prepared and brought.

The significance of the miracle is that it demonstrated that God's presence still dwelt in the Holy Temple.

The Torah Sages instituted the festival of Hanukkah at that time (Talmud, Shabbat 21b), to publicize the miracle (Rashi commentary, ibid). The eight-day rededication of the Temple is also mentioned in the book of Maccabees (I, 4:36; and II, 1:18); and Josephus mentions the eight-day festival in Antiquities ch.12.

The Al-Hanisim prayer which we recite during Hanukkah centers around the Hasmoneans' victory, while the candle-lighting commemorates the miracle of the oil.