The most important Mitzvah of Rosh haShana is listening to the Shofar.
The Torah does not explain the reasons for the Shofar but our rabbis did.
1. The Shofar was used in ancient Israel for the King's coronation. In Rosh haShana we state that God is our King ('ol malkhut shamayim), we are His subjects and follow His rules. The King had the power to sentence or spare someone's life. In Rosh haShana we realize that our lives are evaluated by God, the supreme King/Judge. The key word added/emphasized in the prayers of Rosh HaShana and Yamim Noraim is: King/judge.
2. The Shofar also reminds us of akedat Yitzchak. Abraham Abinu was ready to sacrifice his own son to follow God's commandment. At the end, God told him to hold back because the test was over. Abraham saw a ram caught by its horns in a thicket and he offered it as a sacrifice to God Almighty.
3. The Shofar was also used as a siren, alerting people the enemy was about to attack and their lives were in danger. In Rosh haShana the Shofar serves as a spiritual 'alert call' inviting us to introspect on the fragility of our existence and repent. The Shofar is a wake up call urging us to plead with haShem and request His mercy when judging us.
A modern narrative of akedat Yitzchak and the Shofar: http://www.youtube.com/watch?v=vEOya0ZG0I0