The sounds of the shofar symbolize our crying out to G-d. The different sounds imitate different kinds of crying:
The shevarim (three medium-sized blasts) is a sighing tone, similar to how one would moan when he has great worries.
The Teruah (nine very short blasts) is a wailing sound, similar to how would ululate in a tragic situation.
Before and after these broken blasts, we blow a simple, unbroken blast, called a tekiyah. Chassidic teachings explains that this blast comes from the essence of our soul, and therefore is not composed of multiple parts.