Hebrew, then English. Open "What it means" under any line for the explanation.
Straight on from Hodu, with no break:
רוֹמְמוּ יְיָ אֱלֹהֵינוּ וְהִשְׁתַּחֲווּ לַהֲדֹם רַגְלָיו קָדוֹשׁ הוּא.
Lift up Hashem our God and bow down at His footstool. He is holy.
What it means
A footstool is the low stool where a king rests his feet. Metzudat David (a line-by-line commentary on the Prophets and Writings, from the 1700s) says it is a picture for the Temple: as if God, from His seat in heaven, lowers His feet to rest there.
Metzudat David on Tehillim 99:5רוֹמְמוּ יְיָ אֱלֹהֵינוּ וְהִשְׁתַּחֲווּ לְהַר קָדְשׁוֹ כִּי קָדוֹשׁ יְיָ אֱלֹהֵינוּ.
Lift up Hashem our God and bow down toward His holy mountain, because Hashem our God is holy.
What it means
The holy mountain is Mount Moriah, where the Temple stood (Metzudat David). You will say these two verses again when the Torah is taken out of the ark. The leader calls, "Declare Hashem great with me," and both verses are part of the congregation's answer. The Tur (a law code from the 1300s) already records the custom.
Metzudat David on Tehillim 99:9; Tur, Orach Chaim 134וְהוּא רַחוּם יְכַפֵּר עָוֹן וְלֹא יַשְׁחִית וְהִרְבָּה לְהָשִׁיב אַפּוֹ וְלֹא יָעִיר כָּל חֲמָתוֹ.
And He, the compassionate One, forgives wrongdoing and does not destroy. Again and again He turns back His anger, and He does not wake up all His rage.
What it means
In Tehillim 78 this verse is about the generation of the desert. The verses just before it say they flattered God with their mouths while their hearts were not with Him. He forgave them anyway. It is also the first line of the evening prayer on weeknights.
Tehillim 78:36-38, with Radak; Tur, Orach Chaim 237אַתָּה יְיָ לֹא תִכְלָא רַחֲמֶיךָ מִמֶּנִּי חַסְדְּךָ וַאֲמִתְּךָ תָּמִיד יִצְּרוּנִי.
You, Hashem, will not hold back Your compassion from me. Your kindness and Your truth will always guard me.
What it means
Radak (Rabbi David Kimchi, Provence, about 1200) hears David saying this right after being saved from a danger: You got me out of this one, so You will not hold back Your compassion the next time I need it. And I always need Your kindness and Your truth.
Radak on Tehillim 40:12זְכֹר רַחֲמֶיךָ יְיָ וַחֲסָדֶיךָ כִּי מֵעוֹלָם הֵמָּה.
Remember Your compassion, Hashem, and Your kindnesses, for they are as old as time.
What it means
How far back? Radak makes it personal: back to when I was formed. Your compassion and kindness were with me in my mother's womb, and as I grew up after I was born. You raised me and brought me this far, so it is right for You to have pity on me. I am the work of Your hands. The Story section has the answer of Rashi (the classic commentator, France, died 1105).
Radak on Tehillim 25:6תְּנוּ עֹז לֵאלֹהִים עַל יִשְׂרָאֵל גַּאֲוָתוֹ וְעֻזּוֹ בַּשְּׁחָקִים.
Give strength to God. His majesty is over Israel, and His strength is in the skies.
What it means
How do you give strength to God? Radak says: with words. Say that the strength was His, because it was His strength that beat your enemies, not your own power.
Radak on Tehillim 68:35נוֹרָא אֱלֹהִים מִמִּקְדָּשֶׁיךָ אֵל יִשְׂרָאֵל הוּא נֹתֵן עֹז וְתַעֲצֻמוֹת לָעָם בָּרוּךְ אֱלֹהִים.
You are awesome, God, from Your holy places. The God of Israel, He gives strength and power to the people. Blessed is God.
What it means
The plain sense, in Radak: awe of God reaches people from the Temple, where His presence rests. The Gemara (the rabbis' discussions built on the Mishnah) reads one word differently. Not "from Your holy places" but "from Your holy ones": when God judges His own holy ones, He is feared, raised up and praised.
Radak on Tehillim 68:36; Zevachim 115bאֵל נְקָמוֹת יְיָ אֵל נְקָמוֹת הוֹפִיעַ. הִנָּשֵׂא שֹׁפֵט הָאָרֶץ הָשֵׁב גְּמוּל עַל גֵּאִים.
God of vengeance, Hashem, God of vengeance, show Yourself. Rise up, Judge of the earth. Pay back the arrogant what they deserve.
What it means
This is the opening of Tehillim 94, which the Levites sang in the Temple every Wednesday. Seder HaYom (a guide to the daily prayers, 1599) says verses like this were added after Hodu as a plea for rescue and redemption: that God's rule be over everything and wrongdoing shut its mouth.
Mishnah Tamid 7:4; Seder HaYom, Order of Morning Blessingsלַייָ הַיְשׁוּעָה עַל עַמְּךָ בִרְכָתֶךָ סֶּלָה.
Rescue belongs to Hashem. Your blessing is upon Your people. Selah.
What it means
This is the last line of Tehillim 3, which David wrote while running from his own son Avshalom. Metzudat David points out who "Your people" includes: the people chasing him. David asks to be saved. He does not ask for them to fall.
Tehillim 3:1; Metzudat David on Tehillim 3:9יְיָ צְבָאוֹת עִמָּנוּ מִשְׂגָּב לָנוּ אֱלֹהֵי יַעֲקֹב סֶלָה.
Hashem of hosts is with us. The God of Yaakov is a stronghold for us. Selah.
What it means
In the Talmud Yerushalmi (the Gemara of the Land of Israel), Rabbi Yochanan (the leading rabbi in the Land of Israel in the 200s) says: this verse should never leave your mouth. A second tradition there, in the name of Rabbi Yochanan and his colleagues, names the next verse.
Talmud Yerushalmi, Berakhot 5:1יְיָ צְבָאוֹת אַשְׁרֵי אָדָם בֹּטֵחַ בָּךְ.
Hashem of hosts, happy is the person who trusts in You.
What it means
This is the verse the second tradition in the Yerushalmi names. Radak reads it as a line for a long exile: happy is the one who trusts in You and does not let his heart give up because the exile is so long.
Talmud Yerushalmi, Berakhot 5:1; Radak on Tehillim 84:13יְיָ הוֹשִׁיעָה הַמֶּלֶךְ יַעֲנֵנוּ בְיוֹם קָרְאֵנוּ.
Hashem, save. May the King answer us on the day we call.
What it means
Radak says this is what they used to shout in battle, and that "the King" here is God. Ibn Ezra (a Spanish-born Bible commentator of the 1100s) knows a reading that joins the words: "Hashem, save the king." He rejects it, because the pause in the verse comes right after "save."
Radak and Ibn Ezra on Tehillim 20:10הוֹשִׁיעָה אֶת עַמֶּךָ וּבָרֵךְ אֶת נַחֲלָתֶךָ וּרְעֵם וְנַשְּׂאֵם עַד הָעוֹלָם.
Save Your people and bless Your inheritance. Shepherd them and lift them up forever.
What it means
This verse has exactly ten Hebrew words, and there is a custom to use it to count the ten people of a minyan. The Story section tells why. Radak reads the last words, "forever," as David praying past his own lifetime: my prayer for them is for the days of my life and for after my death, for all the days of the world.
Kitzur Shulchan Aruch 15:3; Radak on Tehillim 28:9נַפְשֵׁנוּ חִכְּתָה לַייָ עֶזְרֵנוּ וּמָגִנֵּנוּ הוּא. כִּי בוֹ יִשְׂמַח לִבֵּנוּ כִּי בְשֵׁם קָדְשׁוֹ בָטָחְנוּ. יְהִי חַסְדְּךָ יְיָ עָלֵינוּ כַּאֲשֶׁר יִחַלְנוּ לָךְ.
Our soul has waited for Hashem. He is our help and our shield. For our heart will be glad in Him, for we have trusted in His holy name. May Your kindness, Hashem, be upon us, just as we have hoped for You.
What it means
These are the last three verses of Tehillim 33. Metzudat David reads the closing words, "just as we have hoped for You," as a measure: the kindness will match the hope.
Tehillim 33:20-22; Metzudat David on Tehillim 33:22הַרְאֵנוּ יְיָ חַסְדֶּךָ וְיֶשְׁעֲךָ תִּתֶּן לָנוּ.
Show us Your kindness, Hashem, and give us Your rescue.
What it means
Radak hears an admission inside this request. Even if our wrongs have piled up, You do kindness with people who have done wrong, and You do not pay them back as their offenses deserve. That is the kindness we are asking to see.
Radak on Tehillim 85:8קוּמָה עֶזְרָתָה לָּנוּ וּפְדֵנוּ לְמַעַן חַסְדֶּךָ.
Rise up and help us, and redeem us for the sake of Your kindness.
What it means
This is the last verse of Tehillim 44. Rashi explains "for the sake of Your kindness": we are not coming on the strength of what we have done. Do it because of Your kindness.
Rashi on Tehillim 44:27אָנֹכִי יְיָ אֱלֹהֶיךָ הַמַּעַלְךָ מֵאֶרֶץ מִצְרָיִם הַרְחֶב פִּיךָ וַאֲמַלְאֵהוּ.
I am Hashem your God, who brought you up from the land of Egypt. Open your mouth wide and I will fill it.
What it means
The one line in this passage where God is the speaker. Rashi: open your mouth wide to ask Me for everything your heart wants, and whatever you ask, I will fill. The Gemara (Berakhot 50a) says this verse was written about words of Torah.
Rashi on Tehillim 81:11; Berakhot 50aאַשְׁרֵי הָעָם שֶׁכָּכָה לּוֹ אַשְׁרֵי הָעָם שֶׁיְיָ אֱלֹהָיו.
Happy is the people for whom it is so. Happy is the people whose God is Hashem.
What it means
In Tehillim 144, "so" points back at a list: children who grow up strong, full storehouses, no cry of alarm in the streets. Radak asks why this people has all that, and answers from the second half of the verse: because Hashem is its God. You will say this verse again as the second line of Ashrei.
Tehillim 144:12-15, with Radakוַאֲנִי בְּחַסְדְּךָ בָטַחְתִּי יָגֵל לִבִּי בִּישׁוּעָתֶךָ אָשִׁירָה לַייָ כִּי גָמַל עָלָי.
As for me, I have trusted in Your kindness. My heart will rejoice in Your rescue. I will sing to Hashem, for He has been good to me.
What it means
This is the last verse of Tehillim 13, a psalm that opens: "How long, Hashem? Will You forget me forever?" Ibn Ezra explains the trust in this line: I have no goodness of my own that I could rely on. I rely only on Your kindness.
Tehillim 13:2; Ibn Ezra on Tehillim 13:6