What does each day of Hanukkah celebrated represent?

Answer 1

The candles we light are in reference to the menorah that stood in the Temple. There is no individual meaning for the different nights.

The reason for all the nights is:

The Seleucids (Syrian-Greeks) under Antiochus Epiphanes (2nd century BCE), at the instigation of the Hellenizers, had forbidden various Torah-practices in Judea, 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 and prepared to light the oil lamps of the Temple's menorah (Exodus ch.25), they found only one day's supply of unsullied olive oil. Miraculously, the menorah stayed lit 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. This is what Hanukkah represents: the closeness to God; and the avoidance of Hellenization (assimilation).

The Torah Sages instituted the festival of Hanukkah at that time (Talmud, Shabbat 21b), to publicize the miracle (Rashi commentary, ibid). This is why we light our Hanukkah-menorahs.

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

It should also be noted that the main goal for which the Maccabees fought was not political independence. They fought to enable the people to observe the Torah's commandments; as we say in the Al Hanisim prayer: "the Greeks sought to cause us to forget Your Torah and leave Your statutes."

Answer 2

The individual days of Hanukkah do not represent anything. But the reason there are 8 days is attributed to this legend:

A story in the Talmud (500 years later) tells of a legendary miracle. After the war, When the Jews returned to rededicate the Ancient Temple, they found that there was only enough oil to keep the eternal lamp burning for 1 day. It would take 8 days to make more of this special oil. So the lit the lamp, and it miraculously burned for 8 days, giving them plenty of time to make more oil to keep the lamp burning.