(כד) דַּבֵּר אֶל בְּנֵי יִשְׂרָאֵל לֵאמֹר בַּחֹדֶשׁ הַשְּׁבִיעִי בְּאֶחָד לַחֹדֶשׁ יִהְיֶה לָכֶם שַׁבָּתוֹן זִכְרוֹן תְּרוּעָה מִקְרָא קֹדֶשׁ. (כה) כָּל מְלֶאכֶת עֲבֹדָה לֹא תַעֲשׂוּ וְהִקְרַבְתֶּם אִשֶּׁה לַיהוָה.
(24) Speak unto the children of Israel, saying: In the seventh month, in the first day of the month, shall be a solemn rest unto you, a memorial proclaimed with the blast of horns, a holy convocation. (25) Ye shall do no manner of servile work; and ye shall bring an offering made by fire unto the LORD.
(א) וּבַחֹדֶשׁ הַשְּׁבִיעִי בְּאֶחָד לַחֹדֶשׁ מִקְרָא קֹדֶשׁ יִהְיֶה לָכֶם כָּל מְלֶאכֶת עֲבֹדָה לֹא תַעֲשׂוּ יוֹם תְּרוּעָה יִהְיֶה לָכֶם.
משנה ראש השנה ג׳:ז׳
(ז) התוקע לתוך הבור או לתוך הדות או לתוך הפיטס, אם קול שופר שמע - יצא, ואם קול הברה שמע - לא יצא. וכן מי שהיה עובר אחורי בית הכנסת, או שהיה ביתו סמוך לבית הכנסת, ושמע קול שופר או קול מגילה, אם כון לבו - יצא, ואם לאו - לא יצא, אף על פי שזה שמע וזה שמע, זה כיוון לבו וזה לא כיוון לבו.
Mishnah Rosh Hashanah 3:7
(7) One who blows into a water cistern, or into a cellar, or into a barrel: if he heard the sound of the shofar, he has fulfilled [his obligation]; if he heard the sound of an echo, he has not fulfilled [his obligation].
And so too, he who was passing behind a synagogue, or whose house was near a synagogue, and heard the sound of a shofar or the sound of [someone reading] the scroll of Esther [on Purim]: if he had intention [to fulfill the commandment], he has performed his duty, but if not, he has not performed his duty. Even though [both] this person heard and that person heard, this one had intention, but that one did not have intention [so they are seen very differently].
(יג) וַיִּשָּׂ֨א אַבְרָהָ֜ם אֶת־עֵינָ֗יו וַיַּרְא֙ וְהִנֵּה־אַ֔יִל אַחַ֕ר נֶאֱחַ֥ז בַּסְּבַ֖ךְ בְּקַרְנָ֑יו וַיֵּ֤לֶךְ אַבְרָהָם֙ וַיִּקַּ֣ח אֶת־הָאַ֔יִל וַיַּעֲלֵ֥הוּ לְעֹלָ֖ה תַּ֥חַת בְּנֽוֹ׃
אמר רבי אבהו למה תוקעין בשופר של איל אמר הקדוש ברוך הוא תקעו לפני בשופר של איל כדי שאזכור לכם עקידת יצחק בן אברהם ומעלה אני עליכם כאילו עקדתם עצמכם לפני
Rabbi Abahu said: Why do we blow a shofar made from a ram's horn? The Holy One, who is blessed, said, "Blow in My presence a shofar made from a ram's horn, and I shall recall for your sake the binding of Isaac, son of Abraham. And I will give you credit as if you had bound yourselves before Me."
Bereishit Rabbah 56:13
א"ר חנינא ב"ר יצחק כל ימות השנה ישראל נאחזים בעבירות ומסתבכין בצרות, ובראש השנה הן נוטלין שופר ותוקעין בו ונזכרים לפני הקב"ה והוא מוחל להם, וסופן ליגאל בקרניו של איל, שנאמר וה' אלקים בשופר יתקע
Genesis Rabbah 56:13
Rabbi Chanina, the son of Rabbi Yitzhak, said: All year long the people Israel are caught in sin and found in the thick of troubles. But on Rosh Hashana they pick up the shofar and blow it, and they are remembered by God who forgives them. And ultimately they will be redeemed with the horns of a ram, as it is stated, "And the Lord God will blow a shofar..." (Zechariah 9:14).
(א) וַיְדַבֵּ֥ר יְהוָ֖ה אֶל־מֹשֶׁ֥ה לֵּאמֹֽר׃ (ב) עֲשֵׂ֣ה לְךָ֗ שְׁתֵּי֙ חֲצֽוֹצְרֹ֣ת כֶּ֔סֶף מִקְשָׁ֖ה תַּעֲשֶׂ֣ה אֹתָ֑ם וְהָי֤וּ לְךָ֙ לְמִקְרָ֣א הָֽעֵדָ֔ה וּלְמַסַּ֖ע אֶת־הַֽמַּחֲנֽוֹת׃ (ג) וְתָקְע֖וּ בָּהֵ֑ן וְנֽוֹעֲד֤וּ אֵלֶ֙יךָ֙ כָּל־הָ֣עֵדָ֔ה אֶל־פֶּ֖תַח אֹ֥הֶל מוֹעֵֽד׃ (ד) וְאִם־בְּאַחַ֖ת יִתְקָ֑עוּ וְנוֹעֲד֤וּ אֵלֶ֙יךָ֙ הַנְּשִׂיאִ֔ים רָאשֵׁ֖י אַלְפֵ֥י יִשְׂרָאֵֽל׃ (ה) וּתְקַעְתֶּ֖ם תְּרוּעָ֑ה וְנָֽסְעוּ֙ הַֽמַּחֲנ֔וֹת הַחֹנִ֖ים קֵֽדְמָה׃ (ו) וּתְקַעְתֶּ֤ם תְּרוּעָה֙ שֵׁנִ֔ית וְנָֽסְעוּ֙ הַֽמַּחֲנ֔וֹת הַחֹנִ֖ים תֵּימָ֑נָה תְּרוּעָ֥ה יִתְקְע֖וּ לְמַסְעֵיהֶֽם׃ (ז) וּבְהַקְהִ֖יל אֶת־הַקָּהָ֑ל תִּתְקְע֖וּ וְלֹ֥א תָרִֽיעוּ׃ (ח) וּבְנֵ֤י אַהֲרֹן֙ הַכֹּ֣הֲנִ֔ים יִתְקְע֖וּ בַּֽחֲצֹצְר֑וֹת וְהָי֥וּ לָכֶ֛ם לְחֻקַּ֥ת עוֹלָ֖ם לְדֹרֹתֵיכֶֽם׃ (ט) וְכִֽי־תָבֹ֨אוּ מִלְחָמָ֜ה בְּאַרְצְכֶ֗ם עַל־הַצַּר֙ הַצֹּרֵ֣ר אֶתְכֶ֔ם וַהֲרֵעֹתֶ֖ם בַּחֲצֹצְר֑וֹת וֲנִזְכַּרְתֶּ֗ם לִפְנֵי֙ יְהוָ֣ה אֱלֹֽהֵיכֶ֔ם וְנוֹשַׁעְתֶּ֖ם מֵאֹיְבֵיכֶֽם׃ (י) וּבְי֨וֹם שִׂמְחַתְכֶ֥ם וּֽבְמוֹעֲדֵיכֶם֮ וּבְרָאשֵׁ֣י חָדְשֵׁיכֶם֒ וּתְקַעְתֶּ֣ם בַּחֲצֹֽצְרֹ֗ת עַ֚ל עֹלֹ֣תֵיכֶ֔ם וְעַ֖ל זִבְחֵ֣י שַׁלְמֵיכֶ֑ם וְהָי֨וּ לָכֶ֤ם לְזִכָּרוֹן֙ לִפְנֵ֣י אֱלֹֽהֵיכֶ֔ם אֲנִ֖י יְהוָ֥ה אֱלֹהֵיכֶֽם׃ (פ)
(1) The LORD spoke to Moses, saying: (2) Have two silver trumpets made; make them of hammered work. They shall serve you to summon the community and to set the divisions in motion. (3) When both are blown in long blasts, the whole community shall assemble before you at the entrance of the Tent of Meeting; (4) and if only one is blown, the chieftains, heads of Israel’s contingents, shall assemble before you. (5) But when you sound short blasts, the divisions encamped on the east shall move forward; (6) and when you sound short blasts a second time, those encamped on the south shall move forward. Thus short blasts shall be blown for setting them in motion, (7) while to convoke the congregation you shall blow long blasts, not short ones. (8) The trumpets shall be blown by Aaron’s sons, the priests; they shall be for you an institution for all time throughout the ages. (9) When you are at war in your land against an aggressor who attacks you, you shall sound short blasts on the trumpets, that you may be remembered before the LORD your God and be delivered from your enemies. (10) And on your joyous occasions—your fixed festivals and new moon days—you shall sound the trumpets over your burnt offerings and your sacrifices of well-being. They shall be a reminder of you before your God: I, the LORD, am your God.
Rambam Hilchot Teshuva 3:4
אע"פ שתקיעת שופר בראש השנה גזירת הכתוב רמז יש בו כלומר עורו ישינים משנתכם ונרדמים הקיצו מתרדמתכם וחפשו במעשיכם וחזרו בתשובה וזכרו בוראכם ... ויעזוב כל אחד מכם דרכו הרעה ומחשבתו אשר לא טוב
Maimonides, Laws of Repentance 3:4
Even though blowing the shofar on Rosh Hashana is a biblical decree [without any explicit purpose given], it does allude to a message, namely: "Wake up, sleepers, from your sleep; and those who are napping, get up from your slumber! Search your deeds, turn in repentance, and recall your Creator...And everyone of you should abandon his wicked ways and his improper plans."
(טז) וַיְהִי֩ בַיּ֨וֹם הַשְּׁלִישִׁ֜י בִּֽהְיֹ֣ת הַבֹּ֗קֶר וַיְהִי֩ קֹלֹ֨ת וּבְרָקִ֜ים וְעָנָ֤ן כָּבֵד֙ עַל־הָהָ֔ר וְקֹ֥ל שֹׁפָ֖ר חָזָ֣ק מְאֹ֑ד וַיֶּחֱרַ֥ד כָּל־הָעָ֖ם אֲשֶׁ֥ר בַּֽמַּחֲנֶֽה׃ (יז) וַיּוֹצֵ֨א מֹשֶׁ֧ה אֶת־הָעָ֛ם לִקְרַ֥את הָֽאֱלֹהִ֖ים מִן־הַֽמַּחֲנֶ֑ה וַיִּֽתְיַצְּב֖וּ בְּתַחְתִּ֥ית הָהָֽר׃ (יח) וְהַ֤ר סִינַי֙ עָשַׁ֣ן כֻּלּ֔וֹ מִ֠פְּנֵי אֲשֶׁ֨ר יָרַ֥ד עָלָ֛יו יְהוָ֖ה בָּאֵ֑שׁ וַיַּ֤עַל עֲשָׁנוֹ֙ כְּעֶ֣שֶׁן הַכִּבְשָׁ֔ן וַיֶּחֱרַ֥ד כָּל־הָהָ֖ר מְאֹֽד׃ (יט) וַיְהִי֙ ק֣וֹל הַשּׁוֹפָ֔ר הוֹלֵ֖ךְ וְחָזֵ֣ק מְאֹ֑ד מֹשֶׁ֣ה יְדַבֵּ֔ר וְהָאֱלֹהִ֖ים יַעֲנֶ֥נּוּ בְקֽוֹל׃ (כ) וַיֵּ֧רֶד יְהוָ֛ה עַל־הַ֥ר סִינַ֖י אֶל־רֹ֣אשׁ הָהָ֑ר וַיִּקְרָ֨א יְהוָ֧ה לְמֹשֶׁ֛ה אֶל־רֹ֥אשׁ הָהָ֖ר וַיַּ֥עַל מֹשֶֽׁה׃
(16) On the third day, as morning dawned, there was thunder [literally: "sounds"], and lightning, and a dense cloud upon the mountain, and a very loud sound of the shofar; and all the people who were in the camp trembled. (17) Moses led the people out of the camp toward God, and they took their places at the foot of the mountain. (18) Now Mount Sinai was all in smoke, for the LORD had come down upon it in fire; the smoke rose like the smoke of a kiln, and the whole mountain trembled violently. (19) The sound of the shofar grew louder and louder. As Moses spoke, God answered him with a sound. (20) The LORD came down upon Mount Sinai, on the top of the mountain, and the LORD called Moses to the top of the mountain and Moses went up.
The Baal Shem Tov, Keter Shem Tov, 194 (adapted from Chabad.org)
A King had an only son, the apple of his eye. The King wanted his son to master different fields of knowledge and to experience various cultures, so he sent him to a far-off country, supplied with a generous quantity of silver and gold. Far away from home, the son squandered all the money until he was left completely destitute. In his distress he resolved to return to his father's house and after much difficulty, he managed to arrive at the gate of the courtyard to his father's palace.
In the passage of time, he had actually forgotten the language of his native country, and he was unable to identify himself to the guards. In utter despair he began to cry out in a loud voice, and the King, who recognized the voice of his son, went out to him and brought him into the house, kissing him and hugging him.
The meaning of the parable: The King is God. The prince is the Jewish people, who are called "Children of God" (Deuteronomy 14:1). The King sends a soul down to this world in order to fulfill the Torah and mitzvot. However, the soul becomes very distant and forgets everything to which it was accustomed to above, and in the long exile it forgets even its own "language." So it utters a simple cry to its Father in Heaven. This is the blowing of the shofar, a cry from deep within, expressing regret for the past and determination for the future. This cry elicits God’s mercies, and He demonstrates His abiding affection for His child and forgives him.