Regarding the holiday of Rosh HaShana, G-d commands us: And in the seventh month, the first day of the month shall be a sacred holiday to you when you may not do any mundane work. It shall be a day of sounding the horn (Numbers 29:1)
The blowing of the shofar, the rams horn, is the significant commandment of Rosh Hashana. But have you ever asked yourself why the Al-mighty saw fit for us to spend the new year in this fashion? What is the significance of the shofar? It is an allusion that all those who are sleeping - who spend their days without examining the consequences of their actions - should arise from their slumber... the voice of the shofar cries out Awake! Examine your deeds; repent and remember your Creator. Those of you who forget the truth and become involved only in vanity and emptiness, look into your souls; improve your ways and actions, forsake your evil path and negative thoughts. (Maimonides, Laws of Repentance, Ch. 3)
One is obligated to listen intently to the sounding of the shofar on Rosh HaShana. He should know that this is the Day of Judgment. As sheep pass before a shepherd who stands and says, this one shall live, and this one shall be slaughtered, so do all the worlds inhabitants pass before G-d. Each ones deeds are inscribed in the Divine record, and read before Him, as it were, and with none knowing whether the verdict had marked him for life or for death. One should therefore listen intently to the shofar, for it reminds man to repent and return to G-d - in order to receive Divine mercy on the Day of Judgment.
The great authority Rav Sadiah Gaon explains that there are ten symbolic reasons for sounding the shofar on Rosh Hashana:
1. This day marks the beginning of creation... for on this day G-d created the world and became its King. And at the beginning of a reign, it is customary to sound trumpets before the newly crowned king, and to proclaim his ascent to sovereignty throughout the realm. Similarly, we accept the Creators sovereignty upon ourselves.
2. Rosh HaShana is the first day of the Ten Days of Repentance, and the shofar is sounded to proclaim and to warn: Whoever wishes to repent, let him repent; if not, he himself will be held responsible.
This is the way of kings; first they forewarn their subjects through decrees, and afterwards, whoever transgresses has no complaint...
3. To remind us of the Revelation, when we stood at Mount Sinai, regarding which the Torah states And the sound of the shofar was exceedingly strong (Ex. 19:19). By hearing the shofar again on this day, we should accept upon ourselves that which our fathers accepted when they said, We will do and we will hear.
4. To remind us of the words of the prophets, which were compared to the sounding of a shofar: And whoever hears the sound of theshofar (i.e., the call of the prophets) and takes no warning - if the sword comes and takes him away, his blood shall be upon his own head; whereas, if he had taken warning, he would have saved his soul (Ezekial 33).
5. To remind us of the destruction of the Holy Temple in Jerusalem and the trumpet-blasting of the enemys attack. When we hear the shofars sound, we are motivated to pray for the renewal of the Temple.
6. To remind us of the binding of Isaac, who offered his life to G-d, and of the ram slaughtered in his place. Likewise, we are to offer our lives for the sanctification of His Name, so that our remembrance may ascend before Him for the good.
7. When we hear the sound of the shofar, we feel trembling and fear, and we humble ourselves before the Creator. For this is the effect of the shofar - it arouses fright and trembling, as it is written: If a shofar is sounded in the city, shall the people not tremble? (Amos 3)
8. To recall in awe the coming great Day of Judgment, as the prophet says: Near is the great day of the L-rd, near and very soon is the day of shofar and shouting (Zephaniah 1).
9. To recall our faith in the future ingathering of the dispersed of Israel, and to awaken our yearning for it. As it is written, And it shall be on that day - a great shofar will be sounded, and those who have perished in the land of Assyria, and those who were dispersed in the land of Egypt will come... (Isaiah 22)
10. To recall our faith in the future resurrection of the dead, as the prophet declares: All you inhabitants of the world, and you who dwell in the earth; when an ensign is lifted on the mountains you shall see, and when theshofar is sounded you shall hear (Isaiah 18).