Chanukah (Hanukkah) begins on the 25th day of the Jewish month of Kislev. Since Hebrew mothns are governed by the moon and, therefore begin near the middle of our own mothns, the holiday can move from perhaps, late November to around the 25th or so of December, by our modern calendar. As with all Jewish feasts and observances, Chanukah actually begins at sundown the evening before the appointed date.
The gift giving depends on the particular branch of Judaism. Some give each night, usuall gelt, (money)either real or chocolate, especially to children. Others save everything until the final night. Actually, gift giving is a relatively recent addition to this celebration, and if you were in Israel during this time, you would find very little emphasis on gifts...