It is customary to blow one hundred and one Shofar sounds on each of the two days of Rosh Hashanah. We blow thirty Shofar sounds before the Musaf service, and then, according to our community’s custom, thirty sounds are blown during the silent Amida prayer of Musaf. Another thirty sounds are blown during the Hazan’s repetition of the Amida, and then, during the Kaddish Titkabal following Musaf, we sound another ten Shofar blasts, bringing the total to one hundred. It is then customary to sound a long "Teru’a Gedola," for a total of one hundred and one.
Why do we blow so many Shofar sounds, beyond that which the Torah strictly requires?
The Gemara raises this question in Masechet Rosh Hashanah (16), and it answers that the additional Shofar sounds serve to "confound the Satan." Rashi (Rabbi Shlomo Ben Yishak, France, 1040-1105) explains that when the Satan sees the Jewish people’s great love for Misvot, to the point where we blow additional Shofar sounds on Rosh Hashanah, he is silenced. His ability to prosecute against us as we stand judgment before God is severely hampered by our display of love and zeal toward the Misvot.
Tosefot (commentaries by the French and German schools of Medieval Talmudists) explain differently, noting that the Satan is also the "Mal’ach Ha’mavet" (angel of death). Based on a comment in the Talmud Yerushalmi, Tosefot write that the angel of death will one day be eliminated, as indicated in the verse (Yeshayahu 25:8), "Bila Ha’mavet La’nesah" ("He shall eliminate death for eternity"). We are also told (Yeshayahu 27:13) that on that day, a great Shofar will be sounded. Thus, when we sound the Shofar after having already fulfilled the Misva of Shofar blowing on Rosh Hashanah, the Satan begins worrying that perhaps this is the Shofar blast that heralds the time of the final redemption, when he will be eliminated. This fear that Satan experiences hinders his ability to prosecute against us in the Heavenly Tribunal.
The Abudarham (Rabbi David Abudarham, 14th century, Spain) cites a Midrash that offers a different explanation of the one hundred Shofar sounds, namely, that it brings to mind the event of Akedat Yishak (the binding of Yishak upon the altar). The Midrash relates that when Sara heard that her son was bound upon the altar prepared to be slaughtered as a sacrifice, she wailed one hundred times. By sounding one hundred Shofar blasts, we bring to mind Sara’s anguish at the time of the Akeda, hoping that in this merit God will atone for our sins and grant us a favorable sentence.
Others explain that the one hundred sounds are reminiscent of the one hundred wailings cried by the mother of the Canaanite general Sisera. The general’s mother wept bitterly one hundred times as she waited in vain for her son to return from his battle against Beneh Yisrael, during which he had been killed. We commemorate her weeping by sounding the Shofar one hundred times on Rosh Hashanah. (At first glance, it seems difficult to understand why Sisera’s mother’s weeping should assume significance on Rosh Hashanah, though this is a subject for a separate discussion.)
We find in Halachic literature some discussion concerning the propriety of the one hundred and first sound that we blow, the "Teru’a Gedola" sounded after the one hundred sounds. The Re’avya (Rabbi Eliezer Ben Yoel Halevi, Germany, 1140-1225), in Siman 541, mentions this custom and expresses his strong disapproval (listen to audio recording for precise citation). He notes that generally speaking, making sounds with an instrument is forbidden on Shabbat and Yom Tob, and sounding the Shofar on Rosh Hashanah is permitted only for the purpose of fulfilling the Halachic obligation. Therefore, once the required Shofar sounds have been blown, it is forbidden to blow the Shofar on Rosh Hashanah any further. A number of works cite the Rosh Yosef as going so far as to claim that blowing a one hundred and first sound constitutes "Hilul Yom Tob" – a desecration of the holiday.
It is clear, however, that these authorities who disapprove of this practice were unaware of the writings of the Geonim, which explicitly record the observance of this custom in the two main Yeshivot of Babylonia, and explain that this, too, serves to confound the Satan. A number of Geonim (specifically Rav Amram Gaon and Rav Hai Gaon) addressed the question of whether the custom was for the one hundred and first sound to be blown publicly or only privately by certain individuals, but, in any event, such a custom most certainly existed. In light of this account, it seems very difficult to reject this custom and consider it a "desecration" of Yom Tob.
Accordingly, Hacham Ovadia Yosef, in his work Hazon Ovadia (Laws of Shofar), codifies this practice, and writes that the one hundred and first sound of the Shofar is blown in order to confound the Satan.
One must ensure, however, not to sound the Shofar after having blown or heard the customary one hundred and one sounds. Of course, if one did not hear all the sounds he may and should certainly blow the sounds he missed, and it is of course permissible to blow the Shofar on behalf of somebody who did not hear the Shofar blowing. One may not, however, blow the Shofar needlessly once he has blown or heard the one hundred and one sounds.
Summary: The accepted custom is to blow one hundred Shofar sounds on Rosh Hashanah, plus an additional "Teru’a Gedola" after these hundred sounds. One may not blow the Shofar unnecessarily once he has heard or blown the one hundred and one sounds.