"The Feast of Trumpets marked the beginning of ten days of consecration and repentance before God. It is one of seven Jewish feasts or festivals appointed by the LORD and one of three feasts that occur in the autumn. The Feast of Trumpets began on the first day (at the new moon) of the seventh month. Its name comes from the command to blow trumpets (Leviticus 23:24; Numbers 29:1-6). It is also called Rosh Hashanah, which means “Head of the Year,” because it marks the beginning of the Jewish civil calendar. During this celebration, no kind of work was to be performed, but burnt offerings and a sin offering were to be brought before the Lord.
Leviticus 23:23 And the LORD spake unto Moses, saying,
Leviticus 23:24 Speak unto the children of Israel, saying, In the seventh month, in the first day of the month, shall ye have a sabbath, a memorial of blowing of trumpets, an holy convocation.
Leviticus 23:25 Ye shall do no servile work therein: but ye shall offer an offering made by fire unto the LORD.
Note: Keep in mind that blood offerings were no longer required after Christ was crucified as the lamb of God as an offering for all sin which means that when observing the Feast of Tabernacles now, no blood offerings or sacrifices are required.
Numbers 29:1 And in the seventh month, on the first day of the month, ye shall have an holy convocation; ye shall do no servile work: it is a day of blowing the trumpets unto you.
What kind of trumpet is used for the Feast of Trumpets:
In the Leviticus passage, the word trumpet is a translation of the Hebrew word hhatsoho-tserah. This designated a straight trumpet, or cornet, in contrast with the shoh-phar, or ram’s horn. But it appears that the shofar was also blown at this time, as it was on the other new moons (Psalm 81:3). Jewish tradition indicates that both kinds of horns were used in the Feast of Trumpets.
Why is the Feast of Trumpets Important?
The Feast of Trumpets was important for several reasons. First, it commemorated the end of the agricultural and festival year. Also, the Day of Atonement fell on the tenth day of this month, and the Festival of Booths began on the fifteenth day. The blowing of the trumpets on first day of the month heralded a solemn time of preparation for the Day of Atonement; this preparation time was called “Ten Days of Repentance” or the “Days of Awe.” The trumpet sound was an alarm of sorts and can be understood as a call to introspection and repentance.
Blowing of Trumpets is also referred to as sounding an alarm for important events such as the Lord's Second Coming.
The Feast of Trumpets, along with the other six festivals of the LORD, foreshadowed certain aspects of the ministry of Jesus Christ. The prophets linked the blowing of trumpets to the future Day of Judgment: “Blow the trumpet in Zion; sound the alarm on my holy hill. Let all who live in the land tremble, for the day of the LORD is coming. It is close at hand” (Joel 2:1; see also Zephaniah 1:14, 16).

In the New Testament, we see that the Lord’s Second Coming will be accompanied by the sound of a trumpet (1 Corinthians 15:51-52; 1 Thessalonians 4:16-17). Each of the judgments in Revelation 8-9 is also signaled by a trumpet. Just as the shofar called the Jewish nation to turn their attention to the Lord and ready themselves for the Day of Atonement, so will the “trump of God” call us to heaven and warn the world of coming judgment" (1).
​
Feast of Trumpets (Rosh Ha-shanah)
The 1st day of the 7th month (Ethanim / Tishri)
Lev 23:24, Num 29:1
A convocation / sabbath day. Not called a feast day. 
No servile work done.
The first day of the Jewish civil year.
This was an announcement to Israel of impending judgment, which occurred on the Day of Atonement, nine days later. The antitype of Trumpets was the worldwide proclamation of the second coming in 1843, during the "Great Awakening" revival, which was based on the 2300 days/years prophecy in Dan 8:14, which began in 457 B.C. and ended in 1844. This was mistakenly interpreted, by William Miller and others, to predict the time of the second coming and end of the world in 1844, when in fact it was the cleansing of the heavenly sanctuary and the beginning of the pre-advent investigative judgment.

---

References:
​1. What is the feast of trumpets? Retrieved on Oct. 2, 2016 from Got questions.org. https://gotquestions.org/Feast-of-Trumpets.html​
2. The Biblical Feast Days. Retrieved on Feb. 29, 2016. http://biblelight.net/feasts.html