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The 13 Attributes
What are the 13 Attributes and where do they come from?
"The core of the Selichot prayers is the 13 Attributes of Mercy, the very words that God taught Moses for the people to use whenever they needed to beg for divine compassion."
–Ronald L. Eisenberg
In Exodus 34, God asks Moses to carve two new tablets to replace the ones he destroyed in anger during the Golden Calf episode. When Moses climbs Mount Sinai with the new tablets, God joins him and makes a speech that contains the 13 Attributes, a sign that God has forgiven Moses and the Israelites for all their mistakes up to that point. Moses further pleads for mercy, and God promises to protect and perform more miracles for the people.
(ו) וַיַּעֲבֹ֨ר יְהֹוָ֥ה ׀ עַל־פָּנָיו֮ וַיִּקְרָא֒ יְהֹוָ֣ה ׀ יְהֹוָ֔ה אֵ֥ל רַח֖וּם וְחַנּ֑וּן אֶ֥רֶךְ אַפַּ֖יִם וְרַב־חֶ֥סֶד וֶאֱמֶֽת׃ (ז) נֹצֵ֥ר חֶ֙סֶד֙ לָאֲלָפִ֔ים נֹשֵׂ֥א עָוֺ֛ן וָפֶ֖שַׁע וְחַטָּאָ֑ה וְנַקֵּה֙ לֹ֣א יְנַקֶּ֔ה פֹּקֵ֣ד ׀ עֲוֺ֣ן אָב֗וֹת עַל־בָּנִים֙ וְעַל־בְּנֵ֣י בָנִ֔ים עַל־שִׁלֵּשִׁ֖ים וְעַל־רִבֵּעִֽים׃

(6) The LORD passed before him and proclaimed: “The LORD! The LORD! a God compassionate and gracious, slow to anger, abounding in kindness and faithfulness, (7) extending kindness to the thousandth generation, forgiving iniquity, transgression, and sin—yet not remitting all punishment, but visiting the iniquity of parents upon children and children’s children, upon the third and fourth generations.”

How do the 13 Attributes break down?

שלש עשרי מידות

יי ׀ יי אֵ֥ל רַח֖וּם וְחַנּ֑וּן אֶ֥רֶךְ אַפַּ֖יִם וְרַב־חֶ֥סֶד וֶאֱמֶֽת ׀ נֹצֵ֥ר חֶ֙סֶד֙ לָאֲלָפִ֔ים נֹשֵׂ֥א עָוֹן וָפֶ֖שַׁע וְחַטָּאָ֑ה וְנַקֵּה֙...וְסָלַחְתָּ֛ לַעֲוֹנֵ֥נוּ וּלְחַטָּאתֵ֖נוּ וּנְחַלְתָּֽנוּ׃

The Thirteen Attributes of God in the Machzor:

The LORD1, The LORD2, a God3 compassionate4 and gracious5, slow to anger6, abundant in kindness7 and truth8, extending kindness to the thousandth generation9, bearing iniquity10, transgression11 and sin12, and cleansing13.... Pardon our iniquity and our sin, and take us for Your own!

The Lord! (Adonai)–God is merciful before a person sins! Even though aware that future evil lies dormant within them.
The Lord! (Adonai)–God is merciful after the sinner has gone astray.
God (El)–a name that denotes power as ruler over nature and humankind, indicating that God’s mercy sometimes surpasses even the degree indicated by this name.
Compassionate (rahum)–God is filled with loving sympathy for human frailty does not put people into situations of extreme temptation, and eases the punishment of the guilty.
Gracious (v’hanun)–God shows mercy even to those who do not deserve it consoling the afflicted and raising up the oppressed.
Slow to anger (ereh apayim)–God gives the sinner ample time to reflect, improve, and repent.
Abundant in Kindness (v’rav hesed)–God is kind toward those who lack personal merits, providing more gifts and blessings than they deserve; if one’s personal behavior is evenly balanced between virtue and sin, God tips the scales of justice toward the good.
Truth (v’emet)–God never reneges on G-d's word to reward those who serve G-d.
Preserver of kindness for thousands of generations (notzeir hesed la-alafim)–God remembers the deeds of the righteous for the benefit of their less virtuous generations of offspring (thus we constantly invoke the merit of the Patriarchs).
Forgiver of iniquity (nosei avon)–God forgives intentional sin resulting from an evil disposition, as long as the sinner repents.
Forgiver of willful sin (pesha)–God allows even those who commit a sin with the malicious intent of rebelling against and angering Him the opportunity to repent.
Forgiver of error (v’hata’ah)–God forgives a sin committed out of carelessness, thoughtlessness, or apathy.
Who cleanses (v’nakeh)–God is merciful, gracious, and forgiving, wiping away the sins of those who truly repent; however, if one does not repent, God does not cleanse.
https://www.myjewishlearning.com/article/the-13-attributes-of-mercy/
Why do we say them during Selichot?
״וַיַּעֲבוֹר ה׳ עַל פָּנָיו וַיִּקְרָא״. אָמַר רַבִּי יוֹחָנָן: אִלְמָלֵא מִקְרָא כָּתוּב, אִי אֶפְשָׁר לְאוֹמְרוֹ. מְלַמֵּד שֶׁנִּתְעַטֵּף הַקָּדוֹשׁ בָּרוּךְ הוּא כִּשְׁלִיחַ צִבּוּר, וְהֶרְאָה לוֹ לְמֹשֶׁה סֵדֶר תְּפִלָּה. אָמַר לוֹ: כׇּל זְמַן שֶׁיִּשְׂרָאֵל חוֹטְאִין — יַעֲשׂוּ לְפָנַי כַּסֵּדֶר הַזֶּה וַאֲנִי מוֹחֵל לָהֶם.
§ The verse states: “And the Lord passed by before him, and proclaimed” (Exodus 34:6). Rabbi Yoḥanan said: Were it not explicitly written in the verse, it would be impossible to say this, as it would be insulting to God’s honor. The verse teaches that the Holy One, Blessed be He, wrapped Himself in a prayer shawl like a prayer leader and showed Moses the structure of the order of the prayer. He said to him: Whenever the Jewish people sin, let them act before Me in accordance with this order. Let the prayer leader wrap himself in a prayer shawl and publicly recite the thirteen attributes of mercy, and I will forgive them.