The information below is an excerpt from The Complete Book of Jewish Observance: A Practical Manual For The Modern Jew by Rabbi Leo Trepp, New York, Behrman House, Inc., 1980, pgs. 94-96.

The Shofar and Its Sounds

The Shofar is a ram's horn, a reminder of the ram offered by Abraham instead of his son Isaac (Gen. 22:13). The horn of a cow or steer may not be used (it might serve as a reminder of the golden calf). The animal from which the horn is to be taken must be kasher. The horn is softened by boiling for several hours; then the cartilage is removed, a hole is drilled into the end that will serve as a mouthpiece, and the hole is then enlarged.

The root of the term Shofar is sh-p-r, hollow. It must, therefore, consist of a perfect, hollow shell, coming to life by the breath of man. No mouthpiece of any material may be added, nor may the Shofar be decorated with any foreign matter, though carvings on the horn itself are permitted. A Shofar should be obtained only from a reliable Jewish dealer and should have a certificate of Kashrut. The Shofar is a symbol of revelation and of redemption. It was sounded at Sinai:

On the third day, as morning dawned, there was thunder and lightning and a dense cloud upon the mountain, and a very loud blast of the Shofar. [Exod. 19:16]

The sound of the Shofar grew louder and louder. As Moses spoke, G-d answered him in thunder. [Exod. 19:19]

It will be heard on the day of Israel's final ingathering. 

And in that day, a great Shofar shall be sounded; and the strayed, who are in the land of Assyria, and the expelled, who are in the land of Egypt, shall come and worship the Lord on the holy mount, in Jerusalem. [Isa. 27:13]

Tradition links the Shofar to the Binding of Isaac, the Akedah, which is read from the Torah on Rosh Hashanah. The ram that Abraham substituted as a sacrifice in Isaac's place had two horns, which G-d preserved. The smaller horn was sounded at Sinai, but the great Shofar will initiate redemption.

The Shofar is also the herald of freedom. By its sound, the year of the Jubilee was initiated, when slaves went free and property was restored to its original owners. We read in Torah: 

Then you shall transmit a blast on the horn; in the seventh month, on the tenth day of the month, the day of Yom Kippur, you shall have the horn sounded throughout the land and you shall hallow the fiftieth year. And proclaim liberty throughout all the land unto all the inhabitants thereof. [Lev. 25:9-10]

The last verse of this Torah section was aptly chosen as the inscription on the American Liberty Bell.

As the blasts on Rosh Hashanah are to be identical to the Shofar sound of the Jubilee, the notes to be sounded have been derived from that Biblical passage. We read in Torah:

Ve-haavarta shofar teruah, Transmit a blast on the horn.

Ve-haavarta, "transmit," signified to the Rabbis a straight, long sound.

Teruah, a blast, must then mean a modified or broken sound. The Rabbis ordained two different forms of broken sound, one a three-break sound, the sigh of a broken heart, and the other a nine-break sound, the whimpering of a weeping soul, and finally both combined. Thus they were sure to have captured the meaning of the injunction. As the term Teruah appears three times in Torah, they decided that the broken blast should be sounded three times, each time preceded by a straight sound and followed by a straight sound.

We call the straight sound Tekiah; the three-break sound Shevarim, and the nine-break sound Teruah. On Rosh Hashanah we sound:

   three times:   Tekiah/Shevarim=Teruah/Tekiah
   three times:   Tekiah/Shevarim/Tekiah
   three times:   Tekiah/Teruah/Tekiah
   stretching the last Tekiah into Tekiah Gedolah, a grand Tekiah.

The three sounds in each group must always be of equal length.