Oikeiōsis is the engine of Stoic ethics. The word literally names the process by which something becomes oikeios — one's own, belonging, familiar. The Stoic doctrine, articulated most fully by Hierocles, is that every living being is naturally constituted to recognize its own existence and welfare as its own, and to pursue them. In humans, who are rational and social, this primal recognition extends outward through reason.
The famous image — Hierocles' concentric circles — places the self at the centre, then the body, then family, then neighbours, then countrymen, then humanity. Each circle is connected to the centre by the same impulse that makes you care for your own body. Ethical development is the work of contracting the outer circles inward — treating distant cases more like near ones.
The doctrine matters because it grounds Stoic kosmopolitēs-citizenship in human nature rather than in arbitrary altruism. It is also the affective counterpart of cosmic sumpatheia — the rational expression, in human social life, of the same mutual-belonging that connects the parts of the cosmos. The Stoic does not love the stranger from charity. He loves the stranger because reason, doing its proper work, recognizes the stranger as continuous with the self by the same impulse that recognizes one's own hand as one's own.
The corollary, often dropped by popular accounts: a Stoicism that flattens out the social dimension and presents the philosophy as solitary resilience training is not Stoicism. It is Stoic technique with the ethics removed. The full doctrine has community at the centre because it has oikeiōsis in the foundation.
Hierocles, Fragment preserved by Stobaeus Anthologium 4.671–673 (concentric circles); cf. Elements of Ethics (papyrus, c. 2nd century CE) for the foundational sunaisthēsis doctrine