The Jewish New Year (Rosh Hashonah)
The Jewish New Year initiates ten days of repentance that culminate with the Day of Atonement. The Torah does not refer to it as Rosh Hashonah (that name first occurs in the Talmud), but as the Memorial of the Blast of the Horn (Zichron Teruah) or the Day of the Blast of the Horn (Yom Teruah). Traditionally, Rosh Hashonah is associated with the Birth of the World.
The Jewish calendar is lunar, which means the 1st day of Tishrei can occur anywhere from the beginning of September to the beginning of October, depending on the year of the Jewish calendar.
Traditional Rosh Hoshonah foods include apples and honey (for a sweet year) and a round challah (egg bread) signifying the cyclicality of a year.
Rosh Hashonah citations in the Torah and Talmud include:
Leviticus 23:24 - Speak to the Children of Israel, saying: in the seventh month on day one of the month there will be for you a Shabbaton [a Sabbath observance], a memorial of the blast of a horn, a solemn assembly.
Numbers 29:1 - And in the seventh month on day one of the month there will be a holy convocation for you; all the work of the service you will not do, it is a day of the blast of the ram's horn for you.
Tractate Rosh Hashonah of the Babylonian Talmud provides the first reference to the 1st day of the 7th month as the New Year. It also connects Rosh Hashonah with the repentance associated with Yom Kippur (the Day of Atonement)
Gedaliah was the governor of Judah, appointed by Nebuchadnezzar, who was assassinated shortly after the fall of the First Temple. This is considered a minor holiday, observed by a fast that begins at sunrise and ends at sunset on the same day (most Jewish holidays begin the night before; this is an exception). The details of Gedaliah's assassination are described in 2 Kings 25:25-26 and Jeremiah 41.