Rosh Hashanah, literally meaning the "head [of] the year", is the Jewish New Year. The biblical name for this holiday is Yom Teruah, literally "day of shouting or blasting". It is the first of the Jewish High Holy Days specified by Leviticus 23:23–32 that occur in the early autumn of the Northern Hemisphere.
The name "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.
There is, however, one important similarity between the Jewish New Year and the American one: Many Americans use the New Year as a time to plan a better life, making "resolutions." Likewise, the Jewish New Year is a time to begin introspection, looking back at the mistakes of the past year and planning the changes to make in the new year, continuing through the Days of Awe and Yom Kippur.
You may notice that the Bible speaks of Rosh Hashanah as occurring on the first day of the seventh month. The first month of the Jewish calendar is Nissan, occurring in March and April. Why, then, does the Jewish "new year" occur in Tishri, the seventh month?
Judaism has several different "new years," a concept which may seem strange at first, but think of it this way: the American "new year" starts in January, but the new "school year" starts in September, and many businesses have "fiscal years" that start at various times of the year. In Judaism, Nissan 1 is the new year for the purpose of counting the reign of kings and months on the calendar, Elul 1 (in August) is the new year for the tithing of animals, Shevat 15(in February) is the new year for trees (determining when first fruits can be eaten, etc.), and Tishri 1 (Rosh Hashanah) is the new year for years (when we increase the year number. Sabbatical and Jubilee years begin at this time).