The phrase now functions in much of Jewish life as the name for social justice as a religious duty. That usage is real and widely held. It is also recent, and it is not what the words meant in either of their earlier careers.
Three distinct things share a root here, and collapsing them manufactures a pedigree.
The legal sense
In the Mishnah the expression appears in a technical setting, as the stated reason for a series of rabbinic enactments: certain rules are adopted mipnei tikkun ha-olam, for the ordering of the world. The rules concern divorce documents, the release of debts, the redemption of captives at a price that will not encourage more kidnapping.
This is legal policy, the smooth functioning of a system and the prevention of perverse incentives. It is not a summons to reform society, and after the talmudic period the formula largely fell out of use.
The liturgical sense
In the Aleinu prayer the worshipper looks toward perfecting the world under divine sovereignty, in a passage whose subject is the ending of idolatry. What is anticipated there is God's doing, prayed for rather than undertaken.
Even this link is less secure than it looks. Some scholars read the verb as establishing rather than repairing, which would detach the prayer from the phrase entirely.
The kabbalistic sense
In the cosmology developed at Safed in the sixteenth century, the vessels meant to hold the divine light shattered, scattering sparks into the material world, and the observance of the commandments with proper intention gathers them.
This is the most dramatic of the three and it points in an unexpected direction: here the repair of the world is accomplished by ritual precision, not by social action. And in the Lurianic texts the phrase usually appears with a specifier, the repair of one of the mystical worlds, rather than of the world.
The modern sense
Its history is not what either side in the argument tends to assume.
The earliest political uses of the phrase, in the 1930s, were European, Orthodox and anti-Zionist, the opposite of its present valence. Its American career appears to begin in the 1950s, with an educator reinterpreting the line from Aleinu as a call to build a just society. The Reform movement's social-justice commitments long predate the phrase and were labelled with it only retrospectively, at the end of the century.
What is actually in dispute
Critics argue that the modern usage lends ancient authority to contemporary politics and converts a demanding tradition into a general ethic. Defenders answer that the prophetic demand for justice is central to Judaism and needs no borrowed phrase to license it, and that earlier modern figures had already linked the term to social ideals.
Both sides accept the philological discontinuity. What they disagree about is whether it is a legitimate development or a reinvention, which is an evaluative question, not a historical one, and this entry does not settle it.