We are obligated to hear 100 shofar blasts on Rosh Hashona [source?] - how are those blasts calculated? Do we actually hear 100 or are there extra?
Dan's answer covered the "how are those blasts calculated?" part of your question; I'll do the other part, "Do we actually hear 100 or are there extra?".
We actually hear 100: 30 initially, 30 in the silent sh'mone esre (nusach S'farad and nusach Ari), 30 in the repetition of the sh'mone esre, and 40 (nusach Ashk'naz) or 10 (nusach S'farad and nusach Ari) in kadish "tiskabel". (I don't know what S'faradim do.)
However, Lubavitchers blow extra after "Alenu" for kabbalistic reasons, bringing the total to 130.
I think that this site http://www.chabad.org/holidays/JewishNewYear/template_cdo/aid/746659/jewish/Why-do-we-blow-the-shofar-so-many-times.htm sums up the computation -- what is important to remember is that a tri-blast like what we call a shevarim is considered "one unit" in the computation as is the truah which is often 9 staccato bursts -- one unit when computing blasts.
Blasts from the Past
The Rabbis developed the rule that there should be 100 blasts during Rosh Hashanah. The issue arises that, if there are nine blasts (biblically) how does one arrive at 100. Thus, prior to the Reader’s Kaddish, 60 blasts should have sounded. If there were nine blasts instead of ten, the count would be 56. See table below.
Shacharis (morning) Service
Torah 30 27
Malchuyot 10 9
Zuchronot 10 9
Shoforot 10 9
Subtotal 60 56
Reader’s Kaddish 40 44
Rabbi Ezra Bick of Yeshivat of Har Etzion, led by R. Soloveichik’s son-in-law, R. Aharon Lichtenstein) states:
There is no REAL significance to the total number. One requires a tekiya-shevarim-tekiya. That unit gets multiplied a few times to reach 9, 9x3 (=30!) or 90 (100). For conventional reasons, we count tekiya-shevarim-terua-tekiya as 4 sounds, although you are correct - it very likely should count as three. But these numbers are purely convention and have no real halakhic significance. We simply call it 30 to keep track.
The Torah requires nine sounds - as in Ccode of Jewish Law (Shulchan Aruch) 590:1-3. The rabbi’s had a problem with the Teruah, whether it is nine short sounds, three medium sounds, or a combination of the two. Therefore, we have three sets of nine; thirty (counting shevarim-teruah as two). The "Schlah" (Siftei Kohen, (1621-1662), a Jewish legal commentator, in The Codeo f Jewish Law (Shulchan Aruch) 592:3-4) was stringent and favored the custom of blowing 30 sounds for "Malchiot, Zichronot and Shofrot" (the three blessings in Musaf) and added ten to complete 100 sounds.
The Shevarim - Teruah, are generally counted as two sounds (in counting the thirty). Halachically, although the Shevarim - Teruah could be counted as one when the congregation is seated, (SA 590:16): however, because the Shevarim and Teruah, are each counted separately in the other two series. the Teruah and Shevarim are counted as two in the counting of thirty, we therefore count them separately. (SA 592:4), where all the hundred sounds are enumerated, and the Teruah and Shevarim are clearly counted as two sounds, in counting the hundred sounds; the custom mentioned in the "Shlah."
Accessed 12/26/01. E-mail by firstname.lastname@example.org (Ezra Bick)
Explanation of the Sound of Shevarim
There are others who point to a commentary in the Talmud, Rosh Hashsnah 33b which points to
Abbaye said: Here there is really a difference of opinion [as to the length of the shevarim compared to the teru’ah]. It is written ‘It shall be a day of treu’ah for you (Num 29:1) and [we translate in Arramaic], a day of yebada, and it is written of the mother of Sisera, ‘Though the window she kooked forth [wa-teyabab]Jud 5:28. One authority thought that this means drawing a long sigh [the one who held that the tru’ah is equal to three shevarim] and the other that is means uttering short piercing cries. referring to blast of the horns.
In Judges 4:22 we find in a battle involving the Prephetess Deborah during the itme of the Jusdges (prior to the establishment of the Israelite Kingdfoms in 1200) defeating the Canaanites in which Jael, another prophetess, entrapped a Canaanite General, Sisera, and killed him.
Behold, as Barak pursued Sisera, Jael came out to meet him, and said to him, "Come, and I will show you the man whom you seek." He came to her; and behold, Sisera lay dead, and the tent peg was in his temples.
Judges 4:22 KJV
To explain Sisera’s Mother and her weeping., the Talmud (Rosh Hashana 33b) indicates that the sound of the teruah as translated as the Targum (Septuagint 130 BCE) translates the phrase as "yom yevava," a day of sobbing, based on the verse (Judges 5:28): "At the window Sisera's mother looked out, and cried." A Midrash (Yal’u’m Shim'oni on Judges 4:3), a codifed rabbinic teaching, tells us that indeed she cried (or sighed) 100 times! http://www.vbm-torah.org/rosh.htm
Issac Klein, former President of the Rabbinical Assembly confirms the 100 blasts:
The Shofar is also sounded at the conclusion of the service to make a total of one hundred blasts (Tosafot, B. R.H. 33b,; Sefer Hamanhig, lii. Rosh Hashanah, sec. 21; Liq. Maharil, p. 180). Customs vary as to how the hundred blasts are completed. Those who follow the Sefardic practice of having thirty blasts during the silent Amidah were shy ten, and added these during the Qaddish Titqabbel that follows the Amidah. Those who follow the Ashkenazic rite were shy forty; they added thirty at Qaddish Titqabbel and ten more after the mourner's Qaddish following Aleinu. The thirty blasts follow the sequence n"vr, n"wn, n"in three times, and the ten follow the same se¬quence once. Each set of thirty is concluded with a long blast—as a sign that the prescribed sounds of this section have come to an end (Maharil).
Isaac Klein, Guide to Jewish Religious Practice, JTS, 1979
The conclusion is that tradition, derived from a rabbinic explanation of a Talmudic gloss, has taken hold in the custom of sounding the shofar 100 times on Tosh HaShanah.