Haunukah is a celebration for what?

Answer 1

Hanukkah is a minor holiday that commemorates the victory of the Jews against the Syrian-Greeks in the Maccabean War of 165 BCE.
The real miracle of Hanukkah is that the Jews were victorious, against all odds, they beat their enemy and were able to restore the Ancient Temple in Jerusalem, which had been desecrated by the Greeks. In a larger context, Hanukkah represents victory against religious persecution and assimilation.
But a second miracle is more commonly known: 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.
This second miracle is very popular with children. It is understood as a legend only, and appears to be the early Rabbis' attempt to add a spiritual level to the holiday.

Answer:
The above answer is non-traditional. Our tradition is that the miracle of the oil did indeed occur and isn't just a legend. If it wasn't for the clear tradition of our Sages, Hanukkah would have been forgotten long ago. Moreover, on Hanukkah the chief celebration is not for the military victory; it's for 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' miraculous military victories (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 in Judea, such as Sabbath-observance and circumcision, rededicated the Temple to a Greek idol, and pressed the Jews to offer up idolatrous sacrifices. Despite being heavily outnumbered, 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. The Hasmoneans (also called Maccabees) were miraculously victorious. When they reached the Temple grounds, they immediately reconsecrated it to God. As part of this action, they relit the Menorah (Exodus ch.25), which was fueled with olive oil. However, it soon became apparent that there was only sufficient oil to keep the candelabra burning for one single day - and it would take eight days to make and bring some more.
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 Hanukkah-menorah, or hanukkiyah, is a special form of the original seven-branched menorah. Our Hanukkah-menorahs have eight spaces for oil, or candles, to mark each of the eight days for which the oil lasted and a ninth to hold the shamash, a candle used to light the others.)
The eight-day rededication of the Temple is also mentioned in pre-Talmudic sources: Megillat Taanit (ch.9), the book of Maccabees (I, 4:56-59; and II, 1:18); and Josephus (Antiquities ch.12).
The Al-Hanisim prayer which we recite during Hanukkah centers around the Hasmoneans' victory and rededication of the Temple, while the candle-lighting commemorates the miracle of the oil.
Though the military victory is prominently mentioned in the prayers, it wouldn't have been celebrated if not for the miracle of the oil, just as we have no special occasion to mark Abraham's victory (Genesis ch.14), or those of Moses (Numbers ch.21), Joshua, Deborah (Judges ch.4), Gideon (Judges ch.6-7), Jephthah (Judges ch.11), or King David. And though the Hasmonean battles continued for two decades after the retaking of the Temple, the Sages instituted Hanukkah immediately after 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."