Definition: Rachamim comes from the Hebrew word Rechem, meaning womb. It is the boundaryless love of a mother for her child, a love that stems from the deep knowledge that the child is both of her and not of her.

God as Compassionate
(ו) וַיַּעֲבֹ֨ר יְהוָ֥ה ׀ עַל־פָּנָיו֮ וַיִּקְרָא֒ יְהוָ֣ה ׀ יְהוָ֔ה אֵ֥ל רַח֖וּם וְחַנּ֑וּן אֶ֥רֶךְ אַפַּ֖יִם וְרַב־חֶ֥סֶד וֶאֱמֶֽת ׀
(6) And the LORD passed by before him, and proclaimed: ‘The LORD, the LORD, God, merciful and gracious, long-suffering, and abundant in goodness and truth;
ה' ה' הוא הממציא שעשה אינו ישנו, והוא המקיים מציאות כל נמצא, שאין לשום נמצא קיום מציאות זולתי השופע ממציאותו:

'ה', ה, a reference to the fact that this is the name of the Lord is the One Who originates matters, called non existent phenomena into existence. The repetition of the name a second time means that it is also He Who is eternal, not subject to fading into the nothingness they came from. At the same time it is He Who preserves these phenomena He has called into existence.

How does this relate to the mother's love?

רחום על החייבים להקל ענשם בקראם אליו, כאמרו פני ה' בעושי רע וכו' צעקו וה' שמע. ורואה בעני נדכאים כענין וגם ראיתי את הלחץ:
רחום, merciful to those who are guilty, reducing the punishment they really deserve. David explains this in Psalms 34,18 צעקו וה' שמו, “they cry out and the Lord hears.” G’d sees and sympathises with the anguish of the downtrodden as we know from Exodus 3,9 וגם ראיתי את הלחץ, “I have also taken note of the oppression, etc.”
Humanity as Compassionate

(ו) כִּ֣י יִקָּרֵ֣א קַן־צִפּ֣וֹר לְפָנֶ֡יךָ בַּדֶּ֜רֶךְ בְּכָל־עֵ֣ץ א֣וֹ עַל־הָאָ֗רֶץ אֶפְרֹחִים֙ א֣וֹ בֵיצִ֔ים וְהָאֵ֤ם רֹבֶ֙צֶת֙ עַל־הָֽאֶפְרֹחִ֔ים א֖וֹ עַל־הַבֵּיצִ֑ים לֹא־תִקַּ֥ח הָאֵ֖ם עַל־הַבָּנִֽים׃ (ז) שַׁלֵּ֤חַ תְּשַׁלַּח֙ אֶת־הָאֵ֔ם וְאֶת־הַבָּנִ֖ים תִּֽקַּֽח־לָ֑ךְ לְמַ֙עַן֙ יִ֣יטַב לָ֔ךְ וְהַאֲרַכְתָּ֖ יָמִֽים׃

(6) If a bird’s nest happens to be before you while you are on your way [walking around], in any tree or on the ground, with young ones or eggs, and the Mother Bird sitting upon the young, or upon the eggs, you shalt not take the Mother Bird with the young; (7) You shall let the Mother Bird go, but the young you may take for yourself [to eat]; In reward: all will be well for you, and that your days will be prolonged [made longer].

From Maimonides Guide of the Perplexed (3, 48)

"People should be restrained and prevented from killing the two together in such a manner that the young is slain in the sight of the mother, for the pain of the animals under such circumstances is very great. There is no difference in this case between the pain of man and pain of other living beings, since the love and tenderness of the mother for her young ones is not produced by reasoning, but by imagination, and this quality (to care for ones' children) exists not only in man but in most living beings."

The Zohar explains that this mitzvah is meant to awaken and intensify God's mercy on All creations. The pain which the mother bird suffers when she is sent away and forced to abandon her young "awakens the forces of mercy in the world" and releases an outpouring of mercy from the heavens above which alleviates all kinds of human suffering.

Here we have added the quality of Rachamim to be something cultivated that is broader than the mother and then God - Creator. It is something that our imagination can create in us towards the whole world.

Compassion through love and through judgement
(כג) וַיְהִי֩ בַיָּמִ֨ים הָֽרַבִּ֜ים הָהֵ֗ם וַיָּ֙מָת֙ מֶ֣לֶךְ מִצְרַ֔יִם וַיֵּאָנְח֧וּ בְנֵֽי־יִשְׂרָאֵ֛ל מִן־הָעֲבֹדָ֖ה וַיִּזְעָ֑קוּ וַתַּ֧עַל שַׁוְעָתָ֛ם אֶל־הָאֱלֹהִ֖ים מִן־הָעֲבֹדָֽה׃ (כד) וַיִּשְׁמַ֥ע אֱלֹהִ֖ים אֶת־נַאֲקָתָ֑ם וַיִּזְכֹּ֤ר אֱלֹהִים֙ אֶת־בְּרִית֔וֹ אֶת־אַבְרָהָ֖ם אֶת־יִצְחָ֥ק וְאֶֽת־יַעֲקֹֽב׃ (כה) וַיַּ֥רְא אֱלֹהִ֖ים אֶת־בְּנֵ֣י יִשְׂרָאֵ֑ל וַיֵּ֖דַע אֱלֹהִֽים׃ (ס)

(23) And it came to pass in the course of those many days that the king of Egypt died; and the children of Israel sighed by reason of the bondage, and they cried, and their cry came up unto God by reason of the bondage. (24) And God heard their groaning, and God remembered His covenant with Abraham, with Isaac, and with Jacob. (25) And God saw the children of Israel, and God took cognizance of them.

The Rabbis see this story as an example of the first type of compassion: compassion from love.

(כג) וַֽיְשַׁלְּחֵ֛הוּ יְהוָ֥ה אֱלֹהִ֖ים מִגַּן־עֵ֑דֶן לַֽעֲבֹד֙ אֶת־הָ֣אֲדָמָ֔ה אֲשֶׁ֥ר לֻקַּ֖ח מִשָּֽׁם׃ (כד) וַיְגָ֖רֶשׁ אֶת־הָֽאָדָ֑ם וַיַּשְׁכֵּן֩ מִקֶּ֨דֶם לְגַן־עֵ֜דֶן אֶת־הַכְּרֻבִ֗ים וְאֵ֨ת לַ֤הַט הַחֶ֙רֶב֙ הַמִּתְהַפֶּ֔כֶת לִשְׁמֹ֕ר אֶת־דֶּ֖רֶךְ עֵ֥ץ הַֽחַיִּֽים׃ (ס)
(23) Therefore the LORD God sent him forth from the garden of Eden, to till the ground from whence he was taken. (24) So He drove out the man; and He placed at the east of the garden of Eden the cherubim, and the flaming sword which turned every way, to keep the way to the tree of life.

This by contrast is an example of compassion through judgement.