Rosh Hashanah.

Rosh Hashanah, is the Jewish New Year. It is the first of the High Holidays or Yamim Noraim ("Days of Awe"), celebrated ten days before Yom Kippur. Rosh Hashanah is observed on the first two days of Tishrei, the seventh month of the Hebrew calendar. It is described in the Torah as  (Yom Teru'ah, a day of sounding [the Shofar]).Laws on the form and use of the shofar and laws related to the religious services during the festival of Rosh Hashanah are described in Rabbinic literature such as the Mishnah that formed the basis of the tractate "Rosh HaShanah" in both the Babylonian Talmud and the Jerusalem Talmud. This also contains the most important rules concerning the calendar year.Rosh Hashanah occurs 163 days after the first day of Passover (Pesach). In terms of the Gregorian calendar, the earliest date on which Rosh Hashanah can fall is September 5, as happened in 1899 and will happen again in 2013. The latest Rosh Hashanah can occur relative to the Gregorian dates is on October 5, as happened in 1967 and will happen again in 2043. After 2089, the differences between the Hebrew calendar and the Gregorian calendar will result in Rosh Hashanah falling no earlier than September 6On Rosh Hashanah day, religious poems, called piyyuttim, are added to the regular services. Special prayer books for Rosh Hashanah and Yom Kippur, called the mahzor (plural mahzorim), have developed over the years. Many poems refer to Psalms 81:4: "Blow the shofar on the [first day of the] month, when the [moon] is covered for our holiday".On the first night of Rosh Hashanah after the evening prayer, it is the Ashkenazi and Hasidic custom to wish Leshana Tova Tikoseiv Vesichoseim (Le'Alter LeChaim Tovim U'Leshalom) which is Hebrew for "May you (immediately) be inscribed and sealed for a Good Year (and for a Good and Peaceful Life)" 
Shana Tova is the traditional greeting on Rosh Hashanah which in Hebrew means "A Good Year.
Shana Tova Umetukah is Hebrew for "A Good and Sweet Year.
Ketiva ve-chatima tovah which translates as "May You Be Written and Sealed for a Good Year.
The formal Sephardic greeting is Tizku leshanim rabbot ("may you merit many years"), to which the answer is ne'imot ve-tovot ("pleasant and good ones"). Less formally, people wish each other "many years" in the local language.