Answer: Well, I don't want to toot the shofar's horn too much, James, but it really is pretty special. Allow me to explain.
In the Torah, we are given a commandment that on the first day of the seventh month (Tishrei), 'you shall observe complete rest, a sacred occasion commemorated with loud blasts.' (Leviticus 23:24) These loud blasts, or teruah, were understood by the rabbis to allude to the blasts of the shofar. So on Rosh Hashanah, we blow the shofar in order to fulfill this commandment. The biblical text doesn't go into precisely the reason that it's so important that we hear a teruah, but there are a few possibilities.
You might imagine that a shofar was chosen for Rosh Hashanah just because it was the only horn-like instrument that the Israelites had in the desert when they were given the commandments. But actually, the Torah mentions a number of instruments the people had with them, including silver trumpets, so the use of the shofar doesn't seem to have been borne from necessity."
