The basic story of what happened when the Hasmonean monarchy overcame the Greeks, and hence we celebrate Chanukah, can be found in the Gemara in Shabbos 21b:
When the Greeks entered the Sanctuary they defiled all the oils that were in the Sanctuary by touching them. And when the Hasmonean monarchy overcame them and emerged victorious over them, they searched and found only one cruse of oil that was placed with the seal of the High Priest, undisturbed by the Greeks. And there was sufficient oil there to light the candelabrum for only one day. A miracle occurred and they lit the candelabrum from it eight days.
The oil used for kindling the menora needed to be "Zach" and "Kosis", e.g. pure and beaten olive oil (See Imrei Binah for an explanation on the quality of oils, e.g. that it flows on top of any other substance).
They could've used impure oil, but since they did not, the Lubavitcher Rebbe explains that this means:
From this we can see the love of Yisrael, who give themselves up even for stringencies and beautification (hidur) of keeping Torah and Mitzvot to the fullest, such as lighting the menorah with specifically pure oil.
In this vein, Rav Dessler explains (Michtav Me'Eliyahu, Chanukah, p. 3):
וכך גם נס פך השמן מה שנתגלה הנס ונקבע זכרו לדורות הוא ללמד כי נקודת אור התורה הפנימית אין לה ביטול לעולם ועתידה לעמוד בפני חשכת ליל ההסתר בכל דור ודור ואף אם ליבו של האדם יתטמטם מ"מ לא תכבה בקרבו נקודת אור התורה הפנימית ודבר זה זכו החשמונאים לגלות מכח מסירות נפשם It comes to teach that the light of the (inner) Torah will never be canceled and will face the darkness of the night in every generation (my own loose translation).
So, according to Rav Dessler, the miracle of Chanukah comes to teach us that the light of Torah (that light was the hidden light of creation that was hidden in the Torah, according to the Baal Shem Tov) will never cease to excist and will never be put out, so to speak.
Similary, the Bnei Yissachar (Kislev/Teves 2:8) explains the meaning of the word Chanukah:
For this reason these days are called Chanukah, from the word chinuch [preparatory training] and practice for what is to come, for the future redemption. [...]
What exactly is the lesson we are meant to take from the Chanukah-story based on the above mentioned sources??