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There are 4 major shofar sounds, or notes, sounded on Rosh Hashana:
What is the Torah source for these specific sounds?
The Torah itself uses only the terms "tekiah" and "teruah" (Num. 10:3ff). Elsewhere (Lev. 25:9) the Torah puts the verb haavir ("to make pass") before and after references to a teruah, implying that it should be preceded and followed by a long drawn-out sound - which tells us that the basic order is tekiah-teruah-tekiah.
So there's no doubt what a tekiah is. However, the word "teruah" is translated in Aramaic as "yevava," which means a crying sound, and there are three possibilities what this means: moaning (medium-length sounds, what we call "shevarim"); sobbing (short sounds, what we call "teruah"); or both, first moaning and then sobbing. Already in the era of the Mishnah there was uncertainty which of these three is the true "teruah"; accordingly, it was instituted that we do all three.
(Talmud, Rosh Hashanah 33b-34a; Shulchan Aruch Harav, Orach Chaim 590:1-2)
[How this uncertainty came about is itself the subject of dispute. R. Hai Gaon explains that in reality, any of these three would satisfy the Torah obligation of hearing a crying sound, so various communities did it in various ways; the enactment to do all three was in order to unify the various practices. By contrast, Rambam writes (Shofar 3:2) that the true original meaning of "teruah" was actually forgotten, and so we do all three to be on the safe side.]
Returning to tekiah, it can be drawn out as long as you want (Shulchan Aruch Harav, Orach Chaim 590:4). So the custom developed to make the last one (of the first set of 30 sounds) extra-long ("tekiah gedolah"), also symbolizing the idea (from Ps. 47:6) that G-d's presence "ascends with the sound of the shofar." But it's not a halachic requirement.
Digging around I found the following dvar torah by Prof. A. Harel Fisch on Bar-Ilan's website regarding a passage from S.Y. Agnon's Days of Awe (Hebrew edition; the English translation is an abridgment) pp. 108-110:
The first round of blowing the shofar, we sound tekiah-shevarim-teru'ah and the last tekiah is not part of this round, but the Rabbis of blessed Memory enacted that one should sound a teruah (succession of tremulous notes), with a simple blast (tekiah) before it and a simple blast after it. The first tekiah is of the essence, the second, a rabbinic enactment. A tekiah is a plain, straight sound, as I have said--the sound that comes out and travels, piercing the air, until it reaches our patriarch Abraham of blessed memory.
Prof. Fisch continues:
Later Agnon characterizes each of the three sounds of the shofar: "The tekiah stands for the totally righteous..." (based on Midrash Yehi Or in Menorat ha-Maor Zohar, also cf. Emor 98). According to this, the plain blast (which precedes the tremulous note) is really essential. This is the simple sound, corresponding to the horn of the wild goat and symbolic of a person whose reasoning is straight and who does not bow over. Next we sound the tremulous note of the person who bends his reason and casts himself down; the latter is symbolized by the curved horn of the ram. These sources cited in Agnon's book support the first tanna mentioned in the Mishnah, and give preference to the tekiah, the simple sound that symbolizes the totally righteous.
If anyone has the book, I'd love to see more.
