The Hebrew calendar month in which Hanukkah usually begins is Kislev.
Hanukkah starts on 25 Kislev, commemorating the rededication of the Second Temple in Jerusalem after the Maccabean victory over the Seleucid Empire. Because the Hebrew calendar is lunisolar, Kislev does not line up with one fixed Gregorian month; Hanukkah can occur from late November through late December.
Hanukkah lasts eight nights, but it may cross into Tevet, the following Hebrew month. Kislev can have either 29 or 30 days, so the holiday may end on 2 or 3 Tevet. This is why Hanukkah’s dates shift from year to year while remaining stable within the Hebrew calendar.
A common mix-up is Tishrei, the month containing Rosh Hashanah and Yom Kippur, or Nisan, the month of Passover. Adar is associated especially with Purim, not Hanukkah. The festival’s date is fixed as 25 Kislev, even though its civil-calendar date changes.