The holiday of Rosh HaShana is the anniversary of the creation of the world and the beginning of the Kingdom of God. On this day Jews reaffirm their acceptance of God as their King.
It celebrates the completion of the Creation which culminated in the creation of Man on the Sixth Day. Hence it is also the birthday of Mankind.
This is the only New Moon which is also a Holy Day. The Day of Trumpets falls on the sabbath of the months, first day of the seventh month, and can never fall on first, fourth and sixth days of the week. The celebrations extends over two days indicating the continuation of creation into history. History started on this day, and renewal and new history of man also will start on this day.
 

Rosh Hashanah is known by several other names:

According to Jewish tradition year after year on this day God judges mankind for the forthcoming year, a judgment that is finally sealed for ten days till the Day of Atonement (Yom Kippur). Hence the traditional greeting on Rosh Hashanah is "May your name be inscribed in the Book of Life for a good year" and on Yom Kippur "May you be sealed in the Book of Life. .

This day in history
(1) Abraham's sacrifice of Isaac in obedience to God led to the promise of redemption (gen 22:18) that through Abraham's "offspring all nations on earth will be blessed"
(2) Samuel the last of the judges was born. His career was the end of the era of judges and the beginning of the eral of Kingship. In the same way the Day of Trumpets brings about radical changes in the history of Israel, Christians and to the rest of mankind.

The forty day period from the first day of Elul through the tenth day of Tishri (Yom Kippur) was to be a time of special spiritual preparation Most Eastern European Jews celebrate as a day of repentance and renewal. They wear white robes and dress even the Torah scrolls and the ark with white curtain and covers instead of the ornate colored ones during the normal period. They take a special water immersion (tevilah mikveh) to purify themselves. Some Groups gather themselves around a gatheirng of water (ocean, lake or stream) and after a service throw bread crumbs away in the water to symbolize thrwoing away sin and impurities. 'You will hurl all our iniquities into the depths of the sea."

They visit the graves of one's departed family members before Rosh Hashanah, during the month of Elul inan affirmation of the hope of resurrection.

The Trumpet - shofar - is sounded daily a 101 times. During the period of temples sacrifices were made continuously. The trumpet is made from ram's horn which in the temple period came from the sacrificial rams. The trumpets were covered with ornamental silver.

Essential use of the trumpet was a call for regathering and preparation. It is more like the bell ringing of the medeaval churches.

"Make two silver trumpets for yourself; you shall make them of hammered work; you shall use them for calling the congregation and for directing the movement of the camps. When they blow both of them, all the congregation shall gather before you at the door of the tabernacle of meeting" Numbers 10:2-3

There are three sounds made with the shofar:

Tekiah - One long blast.
Shevarim - Three shorter blasts.
Teruah - A series of quick blasts.

In the context of the Kingship of Yhvh the three types of sounds in one trumpet blast declare the Godhead as the oneness within the Trinitarian Godhead. The nature of the godhead is also indicated in the blast in their relation with Man. God Almighty (Father - the law giver), God who incarnated (Son - who served mankind), God who is active in the hearts of all who receive him (Holy Spirit - who continues to strive with every man).

In the Sephardi tradition, the number 101 corresponds to the name of the Angel Michael, the protector and deliverer of Israel, of "everyone that shall be found written in the book." (Daniel 12:1). The trumpet call is evidently given by the Arch Angel Michael, which has significance in terms of the final taking away of the church.

Other explanations are

All these points clearly to Prophetic fulfilment in Mesia. A careful study of the life of Jesus leads us to think that Jesus began his ministry on the day of the Trumpets. Just before that he underwent the mikvah of purification in the presence of John the baptist (Matthew 3:13 – 17) following this baptism he went into the wilderness to fast and pray for 40 days and 40 nights (Matthew 4; 1 – 11) Soon after Jesus took over the ministry from all the hands of the previous prophets and began to proclaim, "Repent, the Kingdom of God is at hand." This relevance is again magnified with the association of the day of trumpets with Abraham's sacrifice where Issac was redeemed with a ram

God's Plan began in the First Month of His Calendar (Passover and Feast of Unleavened Bread), continues 50 days later (Pentecost), and will end with the seventh month of His Calendar. The first day of the seventh month (seven is the number for perfection and completion) is the Day of Trumpets which will herald the return of Jesus Christ who will begin another era in the plan of God.

The implication of the trumpets to the resurrection of the dead and the snatching away of the believers by messiah was known to the Jewish scholars. Thus we have in the mystic tradition:

It is this tradition that Paul reiterates:

For the Lord himself will come down from heaven with a rousing cry, with a call from one of the ruling angels, and with God's shofar; those who died united with the Messiah will be the first to rise; then we who are left still alive will be caught up (Latin = rapture) with them in the clouds to meet the Lord in the air; and thus we will always be with the Lord. So encourage each other with these words"
(I Thessalonians 4:16 – 18).

Jesus himself gave the confirmation:

He (the Son of Man) will send out his angels with a great shofar; and they will gather together his chosen people from the four winds, from one end of heaven to the other
(Matthew 24:31).

Matthew 24:31, And He shall send His angels with a great sound of a trumpet, and they shall gather together His elect from the four winds . . . .

I Corinthians 15:52, In a moment, in the twinkling of an eye, at the last trump: for the trumpet shall sound, and the dead shall be raised incorruptible, and we shall be changed.

I Thessalonians 4:16-17, For the Lord Himself shall descend from heaven with a shout, with the voice of the archangel, and with the trump of God: and the dead in Christ shall rise first: Then we which are alive and remain shall be caught up together with them in the clouds, to meet the Lord in the air: and so shall we ever be with the Lord.

While for those who have put their trust in Jesus the trumpet will signify a day of joy for others it will be the beginning of yet another period of tribulation. In Revelation 8, we find seven angels which have seven trumpets ready to sound so as to fulfil the beginning of the Fall Harvest.. Regathering of Israel was also forseen:

"In that day the Lord will thresh from the flowing Euphrates to the Wadi of Egypt, and you, 0 Israelites will be gathered up one by one. And in that day a great trumpet will sound. Those who were perishing in Assyria and those who were exiled in Egypt will come and worship the Lord on the holy mountain in Jerusalem"
(Isaiah 27:12-13).

This call a call of war. It brings about great plagues as each trumpet blast is peeled. The Seventh Plague ends in the battle of Armageddon in Rev. 16:16

Ref:  Festivals of the Jewish Year, By Theodor H. Gaster (New York, 1953).

http://www.jewishheritage.com/resources/lifecycles/rosh_hashana/