Because it’s celebrated during the Christmas season, Hanukah is badly understood. It’s not a Jewish Christmas, or even — strictly speaking — a religious holiday. It was the ancient Israelis’ Fourth of July, and its (true) story was deeply influential in the creation of the United States.
A little background: In the earliest historical (non-biblical) records, the land of Israel — which corresponds to modern Israel, the West Bank, and western Jordan — was divided into two Hebrew kingdoms, Israel in the north and Judea in the south. In 720 b.c., the Kingdom of Israel was conquered by Assyria, and its inhabitants were sold into slavery; these are the so-called “ten lost tribes”; of the original twelve Hebrew tribes, ten lived in Israel, and the other two, Judah and Benjamin, in Judea. (Virtually all modern Jews descend from those two tribes.) After conquering Israel, the Assyrians laid siege to the Judean capital, Jerusalem. The siege failed, and peace terms were negotiated. Instead of being annihilated, Judea became a client state of the Assyrian Empire.
When Assyria began its decline in the seventh century b.c., Judea regained its independence. It stayed independent until 586 b.c., when it was conquered by the Babylonian Empire. In 539 b.c., Babylon fell to the Persian Empire, wherein Judea remained until 332 b.c., when it — along with the rest of the world — was conquered by Alexander the Great.
Alexander had a soft spot for the Jews (possibly because his teacher, Aristotle, had studied with a Jew; possibly because a Jewish legend predicted that a Greek would vanquish the Persian Empire). He annihilated Gaza, but passed through Judea in peace, reportedly stopping in Jerusalem to give an offering at the Temple. Judea became part of the Greek Empire, but the Judeans were allowed to maintain their independent legislative-judicial system, and to continue to practice Judaism.
When Alexander died, his empire was split up by his generals. Judea became the northernmost part of the Ptolemaic (Greek–Egyptian) kingdom until 200 b.c., when it was captured by the (Syrian–Greek) Seleucids. At first, the Seleucids granted the Jews permission to continue to “live according to their ancestral customs,” but that decree was revoked in 175 b.c., when Judea was invaded by King Antiochus IV, who was determined to put it under direct Seleucid control. He captured Jerusalem, banned Judaism, and dedicated the Jewish Temple to Zeus. This is where the story of Hanukah begins.