Chanukah, the Jewish festival of rededication, also known as the festival of lights, is an eight day festival beginning on the 25th day of the Jewish month of Kislev.
Chanukah is probably one of the best known Jewish holidays, not because of any great religious significance, but because of its proximity to Christmas. Many non-Jews (and even many assimilated Jews!) think of this holiday as the Jewish Christmas, adopting many of the Christmas customs, such as elaborate gift-giving and decoration. It is bitterly ironic that this holiday, which has its roots in a revolution against assimilation and suppression of Jewish religion, has become the most assimilated, secular holiday on our calendar.
The story of Chanukah begins in the reign of Alexander the Great. Alexander conquered Syria, Egypt and Palestine, but allowed the lands under his control to continue observing their own religions and retain a certain degree of autonomy. Under this relatively benevolent rule, many Jews assimilated much of Hellenistic culture, adopting the language, the customs, the dress, etc., in much the same way that Jews in America today blend into the secular American society.
More than a century later, a successor of Alexander, Antiochus IV was in control of the region. He began to oppress the Jews severely, placing a Hellenistic priest in the Temple, massacring Jews, prohibiting the practice of the Jewish religion, and desecrating the Temple by requiring the sacrifice of pigs on the altar. Two groups opposed Antiochus: a basically nationalistic group led by Mattathias the Tasmanian and his son Judah Maccabee, and a religious traditionalist group known as the Chasidim, the forerunners of the Pharisees (no direct connection to the modern movement known as Chasidism). They joined forces in a revolt against both the assimilation of the Hellenistic Jews and oppression by the Selucid Greek government. The revolution succeeded and the Temple was rededicated.
According to tradition as recorded in the Talmud, at the time of the rededication, there was very little oil left that had not been defiled by the Greeks. Oil was needed for the menorah (candelabrum) in the Temple, which was supposed to burn throughout the night every night. There was only enough oil to burn for one day, yet miraculously, it burned for eight days. An eight day festival was declared to commemorate this miracle. Note that the holiday commemorates the miracle of the oil, not the military victory: Jews do not glorify war.
Chanukah is not a very important religious holiday. The holiday's religious significance is far less than that of Rosh Hashanah, Yom Kippur, Sukkot, Passover, and Shavu'ot. It is roughly equivalent to Purim in significance, and you won't find many non-Jews who have even heard of Purim! Chanukah is not mentioned in Jewish scripture; the story is related in the book of Maccabbees, which Jews do not accept as scripture.
The only religious observance related to the holiday is the lighting of candles. The candles are arranged in a candelabrum called a menorah that holds nine candles: one for each night, plus a shammus (servant) at a different height. On the first night, one candle is placed at the far right. The shammus candle is lit and three berakhot (blessings) are recited: l'hadlik neir (a general prayer over candles), she-asah nisim (a prayer thanking God for performing miracles for our ancestors at this time), and she-hekhianu (a general prayer thanking G-d for allowing us to reach this time of year). The first candle is then lit using the shammus candle, and the shammus candle is placed in its holder. The candles are allowed to burn out on their own after a minimum of 1/2 hour. Each night, another candle is added from right to left (like the Hebrew language). Candles are lit from left to right (because you pay honor to the newer thing first).
Source: The Jewish Virtual Library