We are bidden by the Torah to sound a Shofar on Rosh Hashanah, as the verse (Bamidbar 29:1) states: And in the seventh month, on the first of the month, it shall be declared a holiday for you, a day of sounding a teruah for you.
Rambam (Hilchot Teshuvah 3) writes:
Although the sounding of the shofar on Rosh Hashanah is a Divine decree [which is accepted without our having to understand its reason], nevertheless, we can discern a purpose in doing so. It is as if it tells us: Sleepers, arise from your slumber, and those who are dozing, awake from your lethargy. Review your actions, repent your sins, and remember your Creator! Those who forget the truth with the passing of time and who waste their years pursuing vanity and folly that is purposeless and cannot save you - look into your souls and improve your ways and your deeds. Let all abandon the ways of evil and thoughts that offer no benefit.
Rav Sa'adyah Gaon writes that the sounding of the shofar on Rosh Hashanah contains ten symbolic elements:
1. Rosh Hashanah marks the beginning of the Creation, the day when G‑d created the world and became its Sovereign. As it is customary to sound trumpets to glorify a king and proclaim one's subservience to him, similarly do we show our acceptance of G‑d's dominion by sounding the shofar.
2. Rosh Hashanah marks the first day of the Ten Days of Repentance. We therefore sound the shofar as a means of announcing and warning that this period has begun. It is as if we announce: Those who choose to repent should do so now, and if they choose not to do so, let them not come later and complain about their fate. This too is the manner in which kings exercise their dominion, announcing their decrees to the accompaniment of trumpet blasts.
3. The sounding of the shofar served as a reminder of the revelation at Mount Sinai, which was also accompanied by shofar blasts. Thus, by listening to the shofar and remembering that event, we once again accept upon ourselves that which our fathers accepted upon themselves.
4. The sounding of the shofar serves to remind us of the remonstrations of our Prophets which are compared to the sounding of the shofar, as the verses (Yechezkel 33:4-5) state: And if the listener shall hear the sound of the shofar and not be careful, then the sword shall come and take him. And if he shall be careful, then his soul has escaped.
5. The sounding of the shofar serves to remind us of the destruction of the Beit ha-Mikdash and the trumpet calls of the armies of our enemies. Thus when we hear the shofar, we should pray for the rebuilding of our Holy Temple.
6. The sounding of the shofar serves as a reminder of the shofar of the ram at the binding of Yitzchak, who offered his life to G‑d but in whose place the ram was sacrificed instead. We too should stand ready to make our lives a sanctification of His Name and we pray that this serve as a source of merit for us.
7. The sounding of the shofar instills a sense of trepidation and fear that leads us to humble ourselves before G‑d, as the verse (Amos 3:6) states: If the shofar is sounded in the city, will people not tremble?
8. The sounding of the shofar serves to remind us of the forthcoming great Day of judgment, as the verses (Tzefanyah 1:14,16) state: The great day of G‑d is near, close and quick [to come].. . is the day of [the sounding of] the shofar and the teruah.
9. The sounding of the shofar serves as a reminder of the future ingathering of the dispersed exiles of Israel and to awaken our yearning for it, of which the verse (Yeshayahu 27:13) states: And it shall be on that day, the great shofar shall be sounded and those who have been lost among Ashur shall come [back].
10. The sounding of the shofar serves to remind us of the resurrection of the dead, as the verse (ibid. 18:3) states: All those inhabitants of the world and those who dwell in the earth, when a sign is lifted upon the mountains you shall see and when the shofar is sounded you shall hear.