Hebrew, then English. Open "What it means" under any line for the explanation.
After the last prayer of the service. Say Aleinu standing:
עָלֵינוּ לְשַׁבֵּחַ לַאֲדוֹן הַכֹּל לָתֵת גְּדֻלָּה לְיוֹצֵר בְּרֵאשִׁית
It is our duty to praise the Master of everything to give greatness to the Shaper of creation
What it means
Aleinu means "on us": it is up to us to praise. The Kol Bo (a medieval collection of laws and customs) says to say it standing, and notes that Aleinu adds up to 166, the same as the Hebrew for "and standing." Abudarham (a Spanish commentator on the prayers, 1300s) reads the opening as "it is our obligation to praise."
Kol Bo 16; Abudarham, Rosh Hashanah Musafשֶׁלֺּא עָשָׂנוּ כְּגוֹיֵי הָאֲרָצוֹת וְלֺא שָׂמָנוּ כְּמִשְׁפְּחוֹת הָאֲדָמָה
that He did not make us like the nations of the lands and did not place us like the families of the earth
What it means
Abudarham says the verb is "made" because he finds no verse about God "making" any other nation, only Israel, as in "Israel rejoices in its Maker" (Tehillim 149:2). The prayer says what it is contrasting two lines later, in the words "for they bow down."
Abudarham, Rosh Hashanah Musaf; Tehillim 149:2שֶׁלֺּא שָׂם חֶלְקֵנוּ כָּהֶם וְגוֹרָלֵנוּ כְּכָל הֲמוֹנָם.
that He did not assign our portion like theirs and our lot like all their crowds
What it means
that He did not set our portion like theirs, or our lot like all their crowds.
Abudarham gives two sources for this line. The first is Yirmeyahu (the prophet Jeremiah) 10:16: "Not like these is the portion of Yaakov." The second: our portion is to keep God's words, as in "Hashem is my portion; I said I would keep Your words" (Tehillim 119:57). That, he says, is not their portion.
Abudarham, Rosh Hashanah Musaf; Yirmeyahu 10:16; Tehillim 119:57שֶׁהֵם מִשְׁתַּחֲוִים לָהֶבֶל וָרִיק וּמִתְפַּלְּלִים אֶל אֵל לֹא יוֹשִׁיעַ,
for they bow down to vanity and emptiness and pray to a god that cannot save
What it means
The wording comes from the Prophets. "Vanity and emptiness" is from Yeshayahu (the prophet Isaiah) 30:7, about Egypt's useless help, and "a god that cannot save" is Yeshayahu 45:20.
Yeshayahu 30:7 and 45:20Pause briefly, then bow as you say "va'anachnu kor'im" (we bow):
וַאֲנַחְנוּ כּוֹרְעִים וּמִשְׁתַּחֲוִים וּמוֹדִים לִפְנֵי מֶלֶךְ מַלְכֵי הַמְּלָכִים הַקָּדוֹשׁ בָּרוּךְ הוּא,
And we bend the knee, bow low, and give thanks before the King of the kings of kings the Holy One, blessed is He
What it means
"King of kings" was a title for human emperors (Ezra 7:12, Daniel 2:37). The Mishnah (the first written collection of Jewish law) gives God the title "King of the kings of kings," with one more "kings" added (Pirkei Avot 3:1).
Ezra 7:12; Daniel 2:37; Pirkei Avot 3:1שֶׁהוּא נוֹטֶה שָׁמַיִם וְיוֹסֵד אָרֶץ, וּמוֹשַׁב יְקָרוֹ בַּשָּׁמַיִם מִמַּעַל, וּשְׁכִינַת עֻזּוֹ בְּגָבְהֵי מְרוֹמִים,
who stretches out the heavens and lays the foundation of the earth and whose glorious seat is in the heavens above and the Presence of whose might is in the highest heights
What it means
who stretches out the heavens and lays the foundation of the earth, whose glorious seat is in the heavens above, and the Presence of whose might is in the highest heights.
These phrases come from verses. "Stretches out the heavens and lays the foundation of the earth" appears in Zecharyah (the prophet Zechariah) 12:1 and in Yeshayahu (the prophet Isaiah) 51:13. Abudarham links "His glorious seat in the heavens" to Tehillim 103:19: "Hashem set His throne in the heavens."
Zecharyah 12:1; Yeshayahu 51:13; Abudarham, Rosh Hashanah Musafהוּא אֱלֺהֵינוּ אֵין עוֹד, אֱמֶת מַלְכֵּנוּ אֶפֶס זוּלָתוֹ
He is our God, there is no other Our King is true nothing exists besides Him
What it means
The Kol Bo says it heard that Yehoshua (Joshua, who led Israel into the Land after Moshe) wrote Aleinu when he captured Jericho, and signed it backward with his early name, Hoshea. The first letters of "Aleinu," "she-lo sam," "va'anachnu kor'im" and "Hu Eloheinu" are ayin, shin, vav, heh: Hoshea in reverse.
Kol Bo 16; Tashbetz Katan 253כַּכָּתוּב בְּתוֹרָתוֹ וְיָדַעְתָּ הַיּוֹם וַהֲשֵׁבֹתָ אֶל לְבָבֶךָ
as it is written in His Torah You shall know today and bring it back to your heart
What it means
The first paragraph talks about God ("He is our God," "His Torah"). The second turns and talks to God ("we hope in You," "Your Torah"). The Kol Bo notes that the first paragraph is in the third person, and says "in His Torah" is the right wording here.
Kol Bo 16כִּי יְיָ הוּא הָאֱלֺהִים בַּשָּׁמַיִם מִמַּעַל וְעַל הָאָרֶץ מִתָּחַת אֵין עוֹד.
that Hashem is God in the heavens above and on the earth below there is no other
The second paragraph:
עַל כֵּן נְקַוֶּה לְךָ יְיָ אֱלֺהֵינוּ לִרְאוֹת מְהֵרָה בְּתִפְאֶרֶת עֻזֶּךָ
Therefore we hope in You Hashem our God to see soon the glory of Your might
What it means
Abudarham reads "therefore" as pointing back to the first paragraph: because God brought us close to serve Him, we hope for His rescue.
Abudarham, Rosh Hashanah Musafלְהַעֲבִיר גִּלּוּלִים מִן הָאָרֶץ וְהָאֱלִילִים כָּרוֹת יִכָּרֵתוּן
to remove idols from the earth and the false gods will be completely cut off
What it means
Abudarham ties these lines to the prophets. "Remove idols from the earth" follows Zecharyah 13:2, "I will cut off the names of the idols from the earth." "The false gods will be cut off" follows Yeshayahu 2:18, "the false gods will vanish completely."
Abudarham, Rosh Hashanah Musaf; Zecharyah 13:2; Yeshayahu 2:18לְתַקֵּן עוֹלָם בְּמַלְכוּת שַׁדַּי וְכָל בְּנֵי בָשָׂר יִקְרְאוּ בִשְׁמֶךָ,
to fix the world under the kingship of Shaddai and all humanity will call on Your name
What it means
Abudarham says that when idolatry is cut off, God's Presence returns to the whole world and the world is repaired. He ties "all humanity will call on Your name" to Tzefaniah (the prophet Zephaniah) 3:9: "I will turn the peoples to a clear language, so they all call in Hashem's name and serve Him shoulder to shoulder."
Abudarham, Rosh Hashanah Musaf; Tzefaniah 3:9לְהַפְנוֹת אֵלֶיךָ כָּל רִשְׁעֵי אָרֶץ,
to turn to You all the wicked of the earth
What it means
Abudarham ties these words to Zecharyah 8:22: "Many peoples and mighty nations will come to seek Hashem."
Abudarham, Rosh Hashanah Musaf; Zecharyah 8:22יַכִּירוּ וְיֵדְעוּ כָּל יוֹשְׁבֵי תֵבֵל כִּי לְךָ תִכְרַע כָּל בֶּרֶךְ תִּשָּׁבַע כָּל לָשׁוֹן.
will recognize and know all who live in the world that to You every knee will bend every tongue will swear loyalty
What it means
All who live in the world will recognize and know that every knee will bend to You, and every tongue will swear loyalty.
Yeshayahu 45 holds these ideas within a few verses. Verse 20 has "a god that cannot save," verse 22 has "Turn to Me, all the ends of the earth, for there is no other," and verse 23 has "to Me every knee will bend, every tongue will swear." In that chapter, God speaks to the survivors of the nations.
Yeshayahu 45:20-23לְפָנֶיךָ יְיָ אֱלֺהֵינוּ יִכְרְעוּ וְיִפֹּלוּ, וְלִכְבוֹד שִׁמְךָ יְקָר יִתֵּנוּ,
In front of You, Hashem our God they will kneel and fall down and to the honor of Your name they will give glory
וִיקַבְּלוּ כֻלָּם אֶת עֹל מַלְכוּתֶךָ, וְתִמְלֺךְ עֲלֵיהֶם מְהֵרָה לְעוֹלָם וָעֶד,
and they will all accept the yoke of Your kingship and You will rule over them soon forever and ever
What it means
"The yoke of the kingdom" is the Sages' language. In the Mishnah, Rabbi Yehoshua ben Korcha says the first paragraph of Shema comes before the second so that a person first takes on the yoke of heaven's kingship, and then the yoke of the commandments. Abudarham notes that Aleinu borrows the phrase.
Mishnah Berakhot 2:2; Abudarham, Rosh Hashanah Musafכִּי הַמַּלְכוּת שֶׁלְּךָ הִיא וּלְעוֹלְמֵי עַד תִּמְלוֹךְ בְּכָבוֹד,
For the kingdom is Yours and for all the ages of eternity You will rule in glory
כַּכָּתוּב בְּתוֹרָתֶךָ יְיָ יִמְלֺךְ לְעֹלָם וָעֶד.
as it is written in Your Torah Hashem will reign forever and ever
What it means
On Rosh Hashanah, Aleinu opens the Musaf section about God as King, called Malchuyot, and ten verses about kingship follow. This verse is the first of them.
Abudarham, Rosh Hashanah Musaf; Shibbolei HaLeket 290וְנֶאֱמַר וְהָיָה יְיָ לְמֶלֶךְ עַל כָּל הָאָרֶץ בַּיּוֹם הַהוּא יִהְיֶה יְיָ אֶחָד וּשְׁמוֹ אֶחָד.
And it is said Hashem will be King over all the earth on that day Hashem will be one and His name one
What it means
The Gemara asks why this verse says Hashem will be one "on that day," when He is one already. The Hashem Melech prayer, earlier in the morning, quotes the same verse and gives the Gemara's two answers.
Pesachim 50aSome add these three verses after Aleinu. This siddur prints them:
אַל תִּירָא מִפַּחַד פִּתְאֹם וּמִשֹּׁאַת רְשָׁעִים כִּי תָבֹא
Do not fear sudden terror or the ruin of the wicked when it comes
What it means
Esther Rabbah (a midrash, the rabbis' teachings on Megillat Esther) says that after Haman's decree was sealed, Mordechai met three schoolchildren, and these were their three verses. The Taz (a commentary on the Shulchan Aruch, the main code of Jewish law) said them daily while the chazzan said Sim Shalom, and says the people of Provence did too.
Esther Rabbah 7:13; Turei Zahav on Orach Chaim 132:2עֻצוּ עֵצָה וְתֻפָר דַּבְּרוּ דָבָר וְלֹא יָקוּם כִּי עִמָּנוּ אֵל.
Make your plan and it will be broken speak your word and it will not stand for God is with us
וְעַד זִקְנָה אֲנִי הוּא וְעַד שֵׂיבָה אֲנִי אֶסְבֹּל.
Even to old age I am the same and even to gray hair I will carry you
What it means
These are God's words. In Yeshayahu 46:3 God speaks to the people "carried by Me from the womb," and then promises to keep carrying them, to old age and to gray hair.
Yeshayahu 46:3-4אֲנִי עָשִׂיתִי וַאֲנִי אֶשָּׂא וַאֲנִי אֶסְבֹּל וַאֲמַלֵּט.
I have made you and I will lift you and I will carry you and I will rescue you