1) 
Rosh Hashanah marks the anniversary of the day God simultaneously created and became Sovereign of the universe. We blow the shofar
since it is customary to sound the trumpets at the coronation of a new king.
2) Just as a king may proclaim a period of amnesty before he punishes wrongdoers, so the shofar blast proclaims, “Whoever wishes to repent- let him do so now. If not, let him not complain later."
3) At Mount Sinai when the Jews accepted the Torah, “the sound of the shofar continually increased and was very great (Exodus 19:19)." On Rosh Hashanah, the shofar reminds us to renew that commitment.
4) The shofar reminds us of the admonitions of the prophets and their calls to repentance as it is stated, “When I bring a sword upon a land…(the sentry) blows the shofar and warns the people (Ezekiel 33:2-3)."
5) The shofar reminds us to pray for the rebuilding of the destroyed Holy Temple, of which the prophet lamented, “I shall not be silent, for the sound of the shofar have you heard, o my soul, the shout of war. Destruction upon destruction has been proclaimed…(Jeremiah 4:19-20)."
6) The shofar, a rams horn, reminds us of Akeidas Yitzchak, the binding of Isaac (Genesis 22), when Abraham sacrificed a ram in lieu of his son. Thus, may our remembrance ascend before Him for the good.
7) The sound of the shofar inspires fear and trembling in the hearts of all who hear it, as the prophet asks: “Can a shofar be blown in a city, and the people not tremble (Amos 3:6)?"
8) The shofar reminds us of the great and awesome Judgment Day of the future, which is described as a day of shofar blowing and shouting (Zephaniah 1:16).
9) The shofar makes us yearn for the ingathering of the exiles, of which it is said, “And it will be on that day, that a great shofar will be blown, and then… they shall prostrate themselves to God on the holy mountain in Jerusalem.”
10) The shofar recalls the resurrection of the dead, which will be accompanied by the sounding of a shofar as it is said, “All inhabitants of the world and dwellers of the earth… you shall hear when a shofar is sounded (Isaiah 18:3)."
Read more about The Holiday Of Rosh Hashana.
Other articles of interest include:
Rabbi Saadiah Gaon
