What is the significance of Chanukah dates?

Answer 1

Hanukkah always starts on the Hebrew calendar date of 25 Kislev, and lasts for eight days. The Hebrew calendar does not line up with the Western calendar because it has a completely different leap year system that can shift holidays each year by 11 to 28 days.
Here are the coinciding secular dates for the upcoming years. The candle lightings begin on the evening BEFORE the first date:

2013: November 27-December 5
2014: December 16-24
2015: December 6-14

Answer:
The significance of the dates of Hanukkah is that it was on these days that the miracle of the oil occurred. Here is the background:
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 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. 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."