Hebrew, then English. Open "What it means" under any line for the explanation.
On a day with no Tachanun (the weekday prayers of pleading), skip this paragraph:
רִבּוֹנוֹ שֶׁל עוֹלָם כְּמוֹ שֶׁכָּבַשׁ אַבְרָהָם אָבִינוּ אֶת רַחֲמָיו לַעֲשׂוֹת רְצוֹנְךָ בְּלֵבָב שָׁלֵם, כֵּן יִכְבְּשׁוּ רַחֲמֶיךָ אֶת כַּעַסְךָ מֵעָלֵינוּ, וְיָגֹלּוּ רַחֲמֶיךָ עַל מִדּוֹתֶיךָ.
Master of the world, just as Avraham our father held down his compassion to do Your will with a whole heart, so may Your compassion hold down Your anger and lift it from us, and may Your compassion roll over Your other traits.
What it means
The Midrash (Bereishit Rabbah, a classic rabbinic collection on Bereishit) gives these words to Avraham. After the Akedah he says to God: "I held down my compassion to do Your will." So when Yitzchak's children sin, remember this and be filled with compassion. The Talmud Yerushalmi (the Gemara of the Land of Israel) has "my impulse," not "my compassion."
Bereishit Rabbah 56:10; Talmud Yerushalmi, Taanit 2:4וְתִתְנַהֵג עִמָּנוּ יְיָ אֱלֹהֵינוּ בְּמִדַּת הַחֶסֶד וּבְמִדַּת הָרַחֲמִים, וּבְטוּבְךָ הַגָּדוֹל יָשׁוּב חֲרוֹן אַפְּךָ מֵעַמְּךָ וּמֵעִירְךָ וּמֵאַרְצְךָ וּמִנַחֲלָתֶךָ.
Deal with us, Hashem our God, with the trait of kindness and the trait of compassion, and in Your great goodness may Your burning anger turn back from Your people, Your city, Your land and Your inheritance.
What it means
The Gemara (the rabbis' discussions built on the Mishnah) asks what God Himself prays. Its answer: "May My compassion hold down My anger, may My compassion roll over My other traits, and may I deal with My children with the trait of compassion." The end of the last line and the start of this one say those words back to Him.
Berakhot 7aוְקַיֶּם לָנוּ יְיָ אֱלֹהֵינוּ אֶת הַדָּבָר שֶׁהִבְטַחְתָּנוּ בְּתוֹרָתֶךָ, עַל יְדֵי משֶׁה עַבְדֶּךָ כָּאָמוּר.
And keep for us, Hashem our God, the word You promised us in Your Torah through Moshe Your servant, as it is said:
What it means
Much of the prayer up to here is also in the Rosh Hashanah Musaf, where it finishes the sentence Zochreinu began before the Akedah. There one verse follows. Here ten quotes from Tanach follow. The "Remember for us the covenant" passage in Selichot (prayers for forgiveness said on fast days) starts with the same first verse and ends with the same last one.
Machzor Rosh Hashanah Sefard, Musaf; Siddur Sefard, Selichot for fast daysוְזָכַרְתִּי אֶת בְּרִיתִי יַעֲקוֹב וְאַף אֶת בְּרִיתִי יִצְחָק, וְאַף אֶת בְּרִיתִי אַבְרָהָם אֶזְכֹּר וְהָאָרֶץ אֶזְכֹּר.
"I will remember My covenant with Yaakov, and also My covenant with Yitzchak, and also My covenant with Avraham I will remember, and the land I will remember."
What it means
Vayikra 26:42. Rashi (the classic commentator, died 1105) asks why the fathers come in reverse: if Yaakov's merit is not enough, Yitzchak is with him, and then Avraham. And why no "I will remember" for Yitzchak? His ashes are seen before God, heaped on the altar. He was never burned. Rashi's source, the Sifra (an early Midrash on Vayikra), says "as if."
Vayikra 26:42; Rashi there; Sifra, Bechukotaiוְנֶאֱמַר, וְאַף גַּם זֹאת בִּהְיוֹתָם בְּאֶרֶץ אֹיְבֵיהֶם לֹא מְאַסְתִּים וְלֹא גְעַלְתִּים לְכַלּוֹתָם לְהָפֵר בְּרִיתִי אִתָּם כִּי אֲנִי יְיָ אֱלֹהֵיהֶם.
And it is said: "And yet, even with all this, when they are in the land of their enemies, I will not reject them and will not loathe them, to finish them off, to break My covenant with them, for I am Hashem their God."
What it means
Vayikra 26:44. An early teaching in the Gemara reads each phrase as a chapter of history. "Not reject them": the days of the Babylonians, when I gave them Daniel. "Not loathe them": the Greeks, when I gave them the Hasmoneans of the Chanukah story. "To finish them off": Haman, when I gave them Mordechai and Esther. "To break My covenant": the Romans.
Vayikra 26:44; Megillah 11aוְנֶאֱמַר וְזָכַרְתִּי לָהֶם בְּרִית רִאשֺׁנִים אֲשֶׁר הוֹצֵאתִי אֹתָם מֵאֶרֶץ מִצְרַיִם, לְעֵינֵי הַגּוֹיִם לִהְיוֹת לָהֶם לֵאלֹהִים, אֲנִי יְיָ.
And it is said: "I will remember for them the covenant of the first ones, whom I took out of the land of Egypt before the eyes of the nations, to be their God. I am Hashem."
What it means
Vayikra 26:45. On Rosh Hashanah, this is the verse the Musaf prayer quotes at this point. Who are "the first ones"? Rashi, from the Sifra: the tribes. The Sifra learns from this verse that the covenant was made not only with the three fathers but with the tribes as well.
Vayikra 26:45; Rashi there; Sifra, Bechukotai; Machzor Rosh Hashanah Sefard, Musafוְנֶאֱמַר וְשָׁב יְיָ אֱלֹהֶיךָ אֶת שְׁבוּתְךָ וְרִחֲמֶךָ, וְשָׁב וְקִבֶּצְךָ מִכָּל הָעַמִּים, אֲשֶׁר הֱפִיצְךָ יְיָ אֱלֹהֶיךָ שָׁמָּה.
And it is said: "Hashem your God will bring back your captives and have compassion on you. He will gather you again from all the peoples where Hashem your God scattered you.
What it means
Devarim 30:3. The Hebrew does not say God will "bring back" your captives. It says He "will return." In the Gemara, Rabbi Shimon bar Yochai (a sage of the Mishnah) learns from this how loved Israel is: wherever they were exiled, God's presence went with them. And when they are redeemed, God comes back with them.
Devarim 30:3; Megillah 29a; Rashi on Devarim 30:3אִם יִהְיֶה נִדַּחֲךָ בִּקְצֵה הַשָּׁמָיִם מִשָּׁם יְקַבֶּצְךָ יְיָ אֱלֹהֶיךָ וּמִשָּׁם יִקָּחֶךָ. וְנֶאֱמַר וֶהֱבִיאֲךָ יְיָ אֱלֹהֶיךָ אֶל הָאָרֶץ אֲשֶׁר יָרְשׁוּ אֲבֹתֶיךָ וִירִשְׁתָּהּ וְהֵיטִבְךָ וְהִרְבְּךָ מֵאֲבֹתֶיךָ.
Even if your outcasts are at the edge of the sky, from there Hashem your God will gather you, and from there He will take you." And it is said: "Hashem your God will bring you to the land that your ancestors possessed, and you will possess it. He will do good to you and make you more numerous than your ancestors."
What it means
Devarim 30:4-5. Rashi gives a second reason the Torah says God Himself "returns": the day the exiles are gathered is so great and so hard that it is as if He must take each person by the hand, from wherever that person is. His proof is Yeshayahu 27:12: "You will be picked up one by one, children of Israel."
Devarim 30:4-5; Rashi on Devarim 30:3וְנֶאֱמַר עַל יְדֵי נְבִיאֶךָ, יְיָ חָנֵנוּ, לְךָ קִוִּינוּ הֱיֵה זְרוֹעָם לַבְּקָרִים, אַף יְשׁוּעָתֵנוּ בְּעֵת צָרָה. וְנֶאֱמַר, וְעֵת צָרָה הִיא לְיַעֲקֹב וּמִמֶּנָּה יִוָּשֵׁעַ.
And it is said through Your prophet: "Hashem, be gracious to us. We have hoped for You. Be their arm every morning, and our rescue in a time of trouble." And it is said: "It is a time of trouble for Yaakov, but he will be saved from it."
What it means
Yeshayahu 33:2, then the end of Yirmiyahu 30:7. In the first verse, Rashi reads "every morning" as day after day. Whose arm is "their arm"? Rashi: people in the hands of a robber. Radak (Rabbi David Kimchi, Provence, about 1200) reads it as ours: every single morning, be our arm against the enemy.
Yeshayahu 33:2; Yirmiyahu 30:7; Rashi and Radak on Yeshayahu 33:2וְנֶאֱמַר בְּכָל צָרָתָם לוֹ צָר וּמַלְאַךְ פָּנָיו הוֹשִׁיעָם, בְּאַהֲבָתוֹ וּבְחֶמְלָתוֹ הוּא גְאָלָם וַיְנַטְּלֵם וַיְנַשְּׂאֵם כָּל יְמֵי עוֹלָם.
And it is said: "In all their trouble, He was troubled, and the angel of His presence saved them. In His love and His compassion He redeemed them. He lifted them and carried them all the days of old."
What it means
Yeshayahu 63:9. One word here is written one way and read another. Read, it is lo with a vav: in all their trouble, it was trouble "to Him." Written, it is lo with an alef: "not." Rashi explains the written form: He did not trouble them as much as their deeds deserved. The Gemara uses this verse to show that lo written with an alef can still mean "to Him."
Yeshayahu 63:9; Rashi there; Sotah 31aוְנֶאֱמַר, מִי אֵל כָּמוֹךָ נֹשֵׂא עָוֺן וְעֹבֵר עַל פֶּשַׁע, לִשְׁאֵרִית נַחֲלָתוֹ לֹא הֶחֱזִיק לָעַד אַפּוֹ, כִּי חָפֵץ חֶסֶד הוּא.
And it is said: "Who is a God like You, who forgives sin and passes over rebellion for the remnant of His inheritance? He does not hold on to His anger forever, because He delights in kindness.
What it means
Michah 7:18. Rava (a leading sage of the Gemara, about 1,700 years ago) read this verse as a deal. "Whose sin does He forgive? One who passes over a wrong done to him." In Rava's words: whoever lets things go, all his own wrongs are let go.
Michah 7:18; Rosh Hashanah 17aיָשׁוּב יְרַחְמֵנוּ יִכְבּוֹשׁ עֲוֺנוֹתֵינוּ, וְתַשְׁלִיךְ בִּמְצוּלוֹת יָם כָּל חַטֹּאתָם. תִּתֵּן אֱמֶת לְיַעֲקֹב חֶסֶד לְאַבְרָהָם אֲשֶׁר נִשְׁבַּעְתָּ לַאֲבוֹתֵינוּ מִימֵי קֶדֶם.
He will again have compassion on us. He will hold down our sins, and You will throw all their sins into the depths of the sea. You will give truth to Yaakov, kindness to Avraham, as You swore to our ancestors in days long ago."
What it means
Michah 7:19-20. You may know these words from Tashlich. The Rema (Rabbi Moshe Isserles, Poland, 1500s), in his notes on the Shulchan Aruch (the main code of Jewish law), records the custom: on Rosh Hashanah people go to a river to say the verse "You will throw into the depths of the sea." Tashlich is the verse's verb: "You will throw."
Michah 7:19-20; Rema, Orach Chaim 583:2וְנֶאֱמַר, וַהֲבִיאוֹתִים אֶל הַר קָדְשִׁי וְשִׂמַּחְתִּים בְּבֵית תְּפִלָּתִי עוֹלוֹתֵיהֶם וְזִבְחֵיהֶם לְרָצוֹן עַל מִזְבְּחִי, כִּי בֵיתִי בֵּית תְּפִלָּה, יִקָּרֵא לְכָל הָעַמִּים.
And it is said: "I will bring them to My holy mountain and make them glad in My house of prayer. Their burnt offerings and their sacrifices will be accepted on My altar, for My house will be called a house of prayer for all the peoples."
What it means
Yeshayahu 56:7. "My house of prayer" is, word for word, "the house of My prayer." The Gemara reads that literally: the verse does not say "their prayer" but "My prayer." From here we learn that God prays. And what does He pray? The words this paragraph began with: "May My compassion hold down My anger."
Yeshayahu 56:7; Berakhot 7a