The beginning of the High Holidays, a time of solemn self examination and judgment. It comes in the fall and lasts two days. It is the beginning of the religious year so everyone wishes a Happy New Year (Leshana Tova).
Traditional Judaism refers to Rosh Hashana as the 'new year' but this is actually inaccurate from a Scriptural point of view. In Scripture, Rosh Hashana (or "Feast of Trumpets/Shofar) occurs in the seventh month, not the first month. It is not the new year at all according to biblical year reckoning.