Rosh Hashanah is the autumnal festival celebrating the start of the Jewish New Year.
Rosh Hashanah occurs on the first and second days of Tishri. In Hebrew, the phrase Rosh Hashanah literally means "head of the year" and thus the holiday is commonly known as the Jewish New Year. This name is somewhat deceptive to those unaffiliated with Jewish practices, however, as there is little similarity between Rosh Hashanah, one of the holiest days in the Jewish year, and the Western, secular interpretation of New Years.
There is, however, one important similarity between the Jewish New Year and the secular one: many people use New Years as a time to make "resolution" and plan to lead a better life. 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. This period of introspection does not end at the conclusion of Rosh HaShanah but actualy stretches for ten days, known commonly as the Days of Awe, until Yom Kippur.
The name "Rosh Hashanah" is not used in the Bible to discuss this holiday. The Bible refers to the holiday as Yom HaZikkaron (the day of remembrance) or Yom T'ruah (the day of the sounding of the shofar). The holiday is instituted in Leviticus 23:24-25.
The shofar is a ram's horn which is blown somewhat like a trumpet. One of the most important observances of this holiday is hearing the sounding of the shofar in the synagogue. A total of 100 notes are sounded each day. There are four different types of shofar notes: tekiah, a 3 second sustained note; shevarim, three 1-second notes rising in tone, teruah, a series of short, staccato notes extending over a period of about 3 seconds; and tekiah gedolah (literally, "big tekiah"), the final blast in a set, which lasts (I think) 10 seconds minimum. The Bible gives no specific reason for this practice. One that has been suggested is that the shofar's sound is a call to repentance. The shofar is not blown if the holiday falls on a Sabbath.
No work is permitted on Rosh Hashanah.
Another popular observance during this holiday is eating apples dipped in honey, a symbol of our wish for a sweet new year. We also dip bread in honey (instead of the usual practice of sprinkling salt on it) at this time of year for the same reason. In addition to dipping an apple honey, we eat round challah bread to symbolize the circle of the life and the cycle of a new year. The challah is also in the shape of a crown because we refer to God as royalty several times throughout the holidays.
Another popular practice of the holiday is Tashlikh ("casting off"). We walk to flowing water, such as a creek or river, on the afternoon of the first day and empty our pockets into the river, symbolically casting off our sins. This practice is not discussed in the Bible, but is a long-standing custom.
Religious services for the holiday focus on the concept of God's sovereignty.
The common greeting at this time is L'shanah tovah ("for a good year"). This is a shortening of "L'shanah tovah tikatev v'taihatem" (or to women, "L'shanah tovah tikatevi v'taihatemi"), which means "May you be inscribed and sealed for a good year." More on that concept at Days of Awe.
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).
Marcia Pravder Mirkin states that, "On Rosh Hashanah we read the tragic and transforming family story of Sarah, Hagar, Abrham, Ishmael, and Isaac. We read it on a holiday that emphasizes teshuvah, turning around, becoming a better person, living a life closer to what God wants from us. How can we experience teshuvah?...teshuvah needs to include empathy, empathetic listening, paying attention, hearing beyond words to the soul and meaning of what is uttered...On Rosh Hashanah we ask God to be empathetic toward us, even tough empathy was so often lacking in ourselves."
See Extra Day of Jewish Holidays for an explanation of why this holiday is celebrated for two days instead of the one specified in the Bible.Rosh Hashanah will begin at sunset on the night preceeding the following days in the American calendar:
Sources: Judaism 101, Marcia Pravder Mirkin, “Hearken to Her Voice: Empathy as Teshuva,” Beginning Arrow.