Rosh Hashanah is the beginning of the year. On that day Adam, the first man, and Eve were created. Although creation began six days before Adam and Eve were created, the 10th day is nonetheless considered the beginning of the world and Rosh Hashanah was set on that day. For mankind is the center of the universe for whom everything was created. Upon his creation, the entire world was completed and G-d's desire in the world came to its fulfillment.
It is called Rosh Hashanah, literally meaning "Head" of the year, and not merely the beginning of the year. As mentioned, the holidays are a reoccurrence of the original event that took place on that day. Just as in the beginning of creation, G-d considered the creation of the world, so too on every Rosh Hashanah, G-d relates to the world with renewed vigor, in a way which He never did before and which forms His relationship to the world for the entire year to come.
As a head contains the life, and is the nerve center of the entire body which is controlled by the brain, so does Rosh Hashanah contain the life and sustenance for the entire year. This is the significance of the judgment on Rosh Hashanah, and that is why the Rosh Hashanah spirit, in Jewish tradition, is that of solemnity. For, pending on our "turning" toward G-d is how G-d decides, on Rosh Hashanah, to relate to us, and consequently sustain and bless us in fulfillment of all our needs.
It is customary to sound the Shofar at the coronation of a King. Similarly, on Rosh Hashanah we "coronate" G-d, so we sound the Shofar.
The sound of the Shofar is to "awaken" people to repent and return to G-d. As it is written, "Shall the shofar be blown in a city, and the people not tremble?" (Amos 3:6) Its message, in the words of Maimonides, is: "Awaken, ye sleepers, from your slumber, and ponder over your deeds; remember your Creator and go back to Him in penitence. Be not of those who miss realities in their pursuit of shadows, and waste their years in seeking after vain things which cannot profit or deliver. Look well to your souls and consider your acts; forsake each his evil way and thoughts, and return to G-d so that He may have mercy upon you!"
The sounding of the Shofar (ram's horn) takes place during the observance of
Rosh Hashanah (Jewish New Year). It consists of four distinct notes; Tekiah
(one long blast), shevarim (three short blasts), teruah (nine
staccato blasts) and tekiah gedolah (one very long blast). In ancient
times, the sound of the Shofar announced the New Moon and was also used to
herald important events. The Jewish philosopher, Maimonides, declared it a call
to repentance, as it remains today.