On Rosh Chodesh we have a special Kurban (sacrifice) וּשְׂעִ֨יר עִזִּ֥ים אֶחָ֛ד לְחַטָּ֖את לַיהוָ֑ה עַל־עֹלַ֧ת  it is seen in the Siddur:

סידור נוסח ספרד ו׳:א׳:י״ט

וִיהִי רָצוֹן מִלְּפָנֶיךָ יְהֹוָה אֱלֹהַי וֵאלֹהֵי אֲבוֹתַי לְמַלֹּאת פְּגִימַת הַלְּבָנָה וְלֹא יִהְיֶה בָּהּ שׁוּם מִעוּט. וִיהִי אוֹר הַלְּבָנָה כְּאוֹר הַחַמָּה וּכְאוֹר שִׁבְעַת יְמֵי בְרֵאשִׁית כְּמו שֶׁהָיְתָה קוֹדֶם מִעוּטָהּ. שֶׁנֶּאֱמַר אֶת שְׁנֵי הַמְּארוֹת הַגְּדוֹלִים:

Siddur, Sefard 6:1:19

And it should be the will before You Hashem our G-d and the G-d of my forefathers, to complete the missing part of the moon, and it should not be incomplete at all. And the light of the moon should be like the light of the sun and like the light of the seven days of creation before it was reduced, as it says "the two great lights."

What does this mean?  What is this Tikkun?

It is written that:

בראשית א׳:ט״ז

טז. וַיַּ֣עַשׂ אֱלֹהִ֔ים אֶת־שְׁנֵ֥י הַמְּאֹרֹ֖ת הַגְּדֹלִ֑ים אֶת־הַמָּא֤וֹר הַגָּדֹל֙ לְמֶמְשֶׁ֣לֶת הַיּ֔וֹם וְאֶת־הַמָּא֤וֹר הַקָּטֹן֙ לְמֶמְשֶׁ֣לֶת הַלַּ֔יְלָה וְאֵ֖ת הַכּוֹכָבִֽים׃

Bereshis 1:16

(16) Hashem made the two great lights, the greater light to dominate the day and the smaller light to dominate the night, and the stars.

Though we see in Gemarah Chullin that it was not so simple and did not start out that way.  As we see from the pasuk it says that there were 2 great luminaries sounding like they were the same size and in the very next line that one is larger than the other.  What is the story?

חולין ס׳ ב:ה׳-ל׳

רבי שמעון בן פזי רמי כתיב (בראשית א, טז) ויעש אלהים את שני המאורות הגדולים וכתיב את המאור הגדול ואת המאור הקטן אמרה ירח לפני הקב"ה רבש"ע אפשר לשני מלכים שישתמשו בכתר אחד אמר לה לכי ומעטי את עצמך ... אמר הקב"ה הביאו כפרה עלי שמיעטתי את הירח והיינו דאמר ר"ש בן לקיש מה נשתנה שעיר של ראש חדש שנאמר בו (במדבר כח, טו) לה' אמר הקב"ה שעיר זה יהא כפרה על שמיעטתי את הירח

Gemarah Chullin 60b:2-4

Rabbi Shimon ben Pazi asked it is written (Bereshis 1:16) "Hashem made the two great lights" and it is also written "the greater light and the smaller light." The moon said before Hashem, "Master of the universe, is it possible for two kings to use one crown ?" Hashem said to the moon, "Go and make yourself smaller"... Hashem said, "Bring an atonement for Me for reducing the size of the moon." And this is consistent with what Raish Lakish said, what is different about the goat offering on Rosh Chodesh that it says about it (Bamidbar 28:15) "for Hashem"? Hashem said that this goat will be an atonement for My reducing the size of the moon.

This Gemara is very puzzling, Hashem needs an atonement? Hashem is perfect, I would like to open this story up with more details.  

The midrash opens the story further showing us that the moon was so to speak insulated, and asked if I made a legitimate statement then why am I being punished? 

Hashem wanted to appease him and offered that unlike the sun he can rule in the day and night.  The Moon was completely not satisfied saying what good is a small lamp in the light of midday? Hashem again tried to console him and said that Israel will count the days of the months by her, without considering the solar year.   They will know their festivals by you, but the moon was not appeased.  Hashem said that the pious sages of Israel will be named after you, (Shemuel the Small, Yaakov the Small, and David the Small light) but the moon was not appeased.  With this Hashem said “bring atonement for Me for I made the moon small”  and the moon was appeased as we see in Bamidbar that this is the only Kurban were it is written; 28:15

וּשְׂעִ֨יר עִזִּ֥ים אֶחָ֛ד לְחַטָּ֖את לַיהוָ֑ה עַל־עֹלַ֧ת הַתָּמִ֛יד יֵעָשֶׂ֖ה וְנִסְכּֽוֹ׃    

And there shall be one goat as a sin offering to Hashem, to be offered in addition to the regular burnt offering and its libation.

Kurban = to is to לכפר To clean or fix the sin and bring back to his status from before the 'sin'

Although this brings a lot of light to the story it also raises more questions. 

Every Kurban it is written “Sair lachatat” and in Rosh Chodesh, it is written Ve Sair Ezim le Chatat Hashem” (a kid goat as a sin offering of Hashem) if it was not written we could not say it, that it was so to speak wrong to make the moon smaller.  And he needs to bring a Kurban to fix it.  

The moon said what he said and Hashem ‘thought’ (if we can say such a thing) that the moon’s intentions were against the sun when he understood his real intentions Hashem wanted to appease the moon.  

[Hashem know everything, this should not in any way make one think that there are things equal to Hashem rather Hashem wishes to teach us a valuable lesson]

Why does Hashem need a Kurban?  Why did Hashem not just make him big again?

It is apparent that Hashem was happy with the moon’s suggestion and wishes him to stay small is so why the Kurban?   This means that on Rosh Chodesh the moon was able to get her request. Since a Kurban means that the situation is fixed. 

What did the moon really want?