Answer:
This celebration commemorates the time when a small guerrilla (surprise attacks) army of Jews defeated the Syrian king Antiochus IV (c. 215-164 BCE), who had taken over Jerusalem and tried to destroy Judaism.
Hanukkah commemorates the Maccabean War of 165 BCE, between the Jews and the Assyrian-Greeks. 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.