Hanukkah (the Festival of Lights, or Feast of Dedication) begins on the 25th of Kislev and lasts eight days. It commemorates the rededication of the Jerusalem Temple following the Maccabean revolt against the Seleucid Empire (ca. 165 BCE) and the miraculous burning of the Temple menorah for eight days on only one day's worth of oil.
Christ Jesus was conceived at Hanukkah. The angel Ga'vri-El (Gabriel) appeared to Miry'am (Mary) during the last part of Kislev, at the time of Hanukkah (John 10:22-23 shows Christ Jesus celebrating this feast). It is historically more accurate to celebrate Christ Jesus entering the world through conception at Hanukkah rather than celebrating His birth at Christmas.
Christmas is NOT the birthday of Christ Jesus. See FAQ 02 — When Was Christ Jesus Born?
Often, the first day of Hanukkah falls very close to Christmas, but the dates shift each year since the Hebrew calendar is lunisolar.