What is Tikkun Olam?

BDB Hebrew English Lexicon:

† [תָּקַן] vb. become straight (NH תִּקֵּן arrange, put right; Ecclus 47:9 תיקן set in order; Assyrian taḳânu, be well ordered, especially Pi. as NH; Aramaic ܬܩܶܢ be established, firm, Pa. ܬܰܩܶܢ, תַּקֵּין fix, arrange, prepare, etc.; Palm. Pa. erect);—
Qal Inf. cstr. לִתְקֹן Ec 1:15 (opp. מְעֻוָּת bent).
Pi. Inf. cstr. לְתַקֵּן 7:13 make straight (opp. עִוָּה); Pf. 3 ms. תִּקֵּן 12:9 put straight, arrange in order (proverbs).

BDB Hebrew English Lexicon:
III. עלם (√ of foll., meaning dub.; cf. NH עוֹלָם (chiefly world, age); MI 7, 10 עלם, Ph. id., Aramaic עָֽלְמָא, ܥܳܠܡܳܐ, Nab. עלם, Palm. עלמא; Arabic عَالَمٌ creation, world, etc.; Ethiopic ዓለም: aevum, saeculum, etc.; according to Thes al. from i. עלם the hidden, cf. Kö^{ii. 1, 87}; Lag^{BN 115} cp. quadril. عألم primitive waters; Ew^{§ 77 a} cp. Ethiopic ዕለት: time [√ ועל Di⁹²³]; Ba^{ZMG xliv (1890), 685} cp. Assyrian ullûti, ullâ [ullânu, remote time], cf. Dl^{HWB 65} Jen^{ZA vii}).

“Tikkun ha'olam is a phrase that drives legal decisions. It reflects an understanding that part of the law's purpose is to create a more just society, rather than a perfect one... Tikkun Ha'olam may be translated and understood as a recalibration of the world, a recognition that the world is out of balance and that legal remedies are needed in order to readjust the world to a better balance. The focus is not so much on the power of an individual to effect change, but rather on the power of law to correct systematic injustice.” Rabbi Jane Kanarek, "What Does Tikkun Olam Actually Mean?" Righteous Indignation: A Jewish Call for Justice (2008).

When we use the phrase Tikkun Olam today, we aren’t usually thinking of legal practice. I believe when we say Tikkun Olam, we are referring to something more individual: an intention of compassion, generosity and loving-kindness. I would argue this is Chesed, righteousness. Chesed is one of the most central acts in the Torah, which teaches us the importance of taking care of everyone. There are countless examples of this in Jewish text. Acts of Chesed are immediate responses to an individual human need. They are responses to men and women on the street who need a sandwich, or a dollar to get them through the day. They are holding the door for the person who has their hands full of groceries and offering someone else a seat on the bus before you sit down.

They are not, however, Tikkun Ha’Olam. Tikkun Ha’Olam is quite different from these immediate acts of Chesed. It is a response to an overhaul of injustice, a sense that law must be modified to create a more balanced society. Chesed reminds us to be compassionate to the people who make up our societies, and Tikkun Ha’Olam teaches us to try and change society itself. True Tikkun Ha’Olam involves systemic change, not isolated deeds. -Ryan Leszner, URJ Camp George Camper, CIT, Staff 2002-2012

https://campgeorge.org/2020/07/29/it-is-time-for-tikkun-olam-just-not-the-kind-you-think/

"Tikkun olam" seems to be being used as a phrase both amorphous but apparently understood at the time, meaning something like "keeping the world in balance."

This is not the only vague-but-oft-quoted epithet. Another one to investigate is "derech eretz." Literally "the way of the land," this usually means something like "common decency."

https://www.sefaria.org/topics/derekh-eretz?tab=sources

Rabbi Tiferet Berenbaum, writing on Alenu (for MyJewishLearning's weekly Recharge essays):

The other day, I saw this Venn diagram where one circle read, “Your unique talents, skills and gifts” and the other circle read, “What breaks your heart most about this world.” The overlapping area stood for, “Where you are now being called to be in service.”

This image illustrates the two stanzas of Aleinu perfectly. Aleinu is situated at the end of services, as we are about to depart from sacred community, and is like a pep-rally to get us motivated and ready to go back out into the world to serve. ...

We are obligated to praise...but how?

The second stanza answers that question. All that we do is in order to le’taken olam b’Malchut Shaddai... to do tikkun olam through Malchut Shaddai (we’ll translate this later). Many people believe that tikkun olam is simply about fixing the world and doing good. But it is tied to a kabbalistic view of creation that imagines the Divine Creator pouring itself into levels of existence. Along the way, the vessels were unable to contain the all-ness of the Divine and they shattered and spread throughout our world. These shards can be found in all places, especially places that appear at first glance to be dank and devoid of holiness. In these places and these moments, we are called on to search out those shards of the original divine vessel which held the all-ness of the One and lift up those “sparks,” or shards, or the residue of holiness that is left from that original break. So, when we consider, “What breaks your heart the most in this world?” we think about the places where the sparks are most hidden. When we use our unique talents and blessings to effect change where we see a great need, we are truly praising G!d. Aleinu reminds us of our abilities and empowers us to step into difficult places and find G!d there and to know that by acting with G!dly purpose in those moments and in those places, we are doing tikkun olam, repairing the world, by strengthening the world to hold and contain more divinity.

Letaken olam b’Malchut Shaddai. What is Malchut Shaddai? Most simply it is translated as the kingdom of G!d. But the essence of Malchut Shaddai is the interconnectedness of all life. It’s the flow that exists between all things, where each element in creation both provides for and takes from the whole. So, we are charged with doing tikkun olam through these unique gifts because we are given these gifts for no other reason than to do tikkun olam. We have what someone else lacks. We need what someone else can give us. We are all connected, so we owe it to the author of creation to reinvest in the system, to manifest the plan.

Tikkun from a mystical perspective:

From https://www.chabad.org/kabbalah/article_cdo/aid/380568/jewish/Shattered-Vessels.htm

... Due to the intensity and exclusivity of the lights and the inability of their vessels to contain them, the vessels of the lower sefirot of Tohu [chaos] shattered and the lights they contained remained above. The fragments of these vessels then fell to lower levels, becoming absorbed into the various worlds below the world of Tohu.

... Thus Tohu was a primordial form of existence that "was created in order to be destroyed, and destroyed in order to be rebuilt" in a superior form (see Mevo L'Chachmat HaKabbala part 2, shaar 6, ch. 7). The order of creation that followed the disintegration of the world of Tohu is called the world of Tikun (literally translated as "rectification" or "restoration"). Regarding Tikun, the Torah states, "And G‑d looked over everything that He had made, and indeed it was very good" (Gen. 1:31).

... Although the sefirot of Tohu shattered and "died," nevertheless, a residue of the lights that were contained in the vessels remained clinging to the fragments of the vessels.

... The shattering of the sefirot of Tohu is not a coincidence, nor does it signify a flaw in the creative process. On the contrary, it serves a very specific and important purpose, which is to bring about a state of separation or partition of the light into distinct qualities and attributes, and thereby introduce diversity and multiplicity into creation, as explained above. In addition, the shattering of the vessels of Tohu allows for the possibility of evil, and gives man the opportunity to choose between good (for which he gains reward) and evil (for which he is punished). Thus G‑d's attributes of chesed and gevura - the attributes from which reward and punishment derive - are revealed in the world9 , which is one of the primary purposes of creation.

Midrash Tanchuma, Mishpatim 2

If a person of learning participates in public affairs and serves as a judge or arbiter, he gives stability to the land....

But if he sits in his home and says to himself, "What have the affairs of society to do with me?... What should I trouble myself with the people's voices of protest? Let my sould dwell in peace! -- if he does this, he overthrows the world.