Why do we sound the Shofar? What is it supposed to accomplish? The Sefer HaChinuch tells us that we must understand the nature of "man", human beings. As 'man' is a creature from the physical realm, he is only aroused and inspired to action by something stirring, something that will cause him to snap out of the ordinary routine. We see this concept in practice at a time of war: In order to stir up the troops, trumpets are sounded, in hope that this arouses and motivates the soldiers to action. On Rosh HaShana, we do the same. We "awaken", by means of the Shofar, all who are to be judged on this day. We try to incite all who have sinned to plead with Hashem and request mercy from Him when judging. Hashem is receptive, as he is gracious, compassionate and forgiving, of those who return to Him with a complete heart. If the sounding of the Shofar has its intended effect, Hashem will graciously accept the repentance of all on Rosh HaShana.
The sound that eminates from and the shape of the Shofar are meant to inspire us as well. The Sefer HaChinuch writes that the Shofar is a reminder that man should strive to break the impulses of his heart which are evil with the sinful cravings of the world. How does the sound of the Shofar accomplsih this? The actual sound emitted from the Shofar is broken; it is not one straight note, but a series of staccato blasts. The broken sound reminds us that we have a job of "breaking" to do as well - the breaking of our evil inclination.
The shape of the Shofar is not straight like a trumpet. Rather, the end of the Shofar is curved and bent. This bent shape is to remind us that we should bend our hearts in subservience to Hashem.