Rosh Hashanah occurs on the first and second days of Tishri. In Hebrew, Rosh Hashanah means, literally, "head of the year" or "first of the year." Rosh Hashanah is commonly known as the Jewish New Year. This name is somewhat deceptive, because there is little similarity between Rosh Hashanah, one of the holiest days of the year, and the American midnight drinking bash and daytime football game.
Yom Kippur is probably the most important holiday of the Jewish year. Many Jews who do not observe any other Jewish custom will refrain from work, fast and/or attend synagogue services on this day. Yom Kippur occurs on the 10th day of Tishri. The holiday is instituted at Leviticus 23:26 ET seq.
The Festival of Sukkot begins on Tishri 15, the fifth day after Yom Kippur. It is quite a drastic transition, from one of the most solemn holidays in our year to one of the most joyous. Sukkot is so unreservedly joyful that it is commonly referred to in Jewish prayer and literature as Z'man Simchateinu Z'mn Simchateinu (in Hebrew), the Season of our Rejoicing.
Tishri 22, the day after the seventh day of Sukkot, is the holiday Shemini Atzeret. In Israel, Shemini Atzeret is also the holiday of Simchat Torah. Outside of Israel, where extra days of holidays are held, only the second day of Shemini Atzeret is Simchat Torah: Shemini Atzeret is Tishri 22 and 23, while Simchat Torah is Tishri 23.
Chanukkah, 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. Chanukkah 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.