Rosh HaShanah, the Jewish New Year, is observed in a way that is very distinct from how the secular New Year is celebrated. Instead of being a day with general merrymaking, it is a day of reflection.
One common trait between the two holidays is the fact that many make New Year's resolutions, but aside from that, they are worlds apart. When one thinks about Rosh HaShanah, the vision and sound of the shofar-blowing certainly occupies a central aspect of the day's observances. The Torah commands us to blow the shofar on Rosh HaShanah day, but does not explain why this mitzvah is associated so closely with this day. For that explanation, we turn to the Talmud.
The mitzvah to hear the shofar on Rosh HaShanah is not as simple as many imagine. It is not just one sound to which we listen. Instead, there is a whole series of sounds. Our custom is to listen to at least 100 sounds of the shofar on each day of the holiday (except if one of the days is Shabbat, in which case we do not blow the shofar at all).
There are three distinct sounds that are formed. They are called tekiah, shevarim and teruah. The tekiah is the single long burst of the shofar, the shevarim are three medium bursts and the teruah is at least nine short bursts.
The nine bursts of the teruah must continue for at least as long as the sound of the tekiah, and that each of the three bursts of the shevarim should equal three teruahs, making the shevarim also as long as the tekiah.
The different types of sounds are meant to represent different kinds of crying. Why do we imitate the sound of crying? We imitate the sound of crying to instill within us a feeling of remorse for our past misdeeds and to move us to seek a closer relationship with G-d.
The shofar-blowing commences during the latter part of the morning service on Rosh HaShanah after the Torah reading. In preparation for the blowing we read some excerpts from Psalms that talk about the shofar and its immense symbolism and power. Throughout the Musaf service we pause to hear the blowing. It is over the course of this portion of the service that three different sets of blowing occur, each with its own theme.
The three themes are called malchiyot (kingship), shofrot (shofars), and zichronot (remembrances). Each of these themes is meant to elicit within us different thoughts and intentions.