Eternal return names two related but genuinely distinct things, and separating them prevents a good deal of confusion.
In the study of religion, following Mircea Eliade's The Myth of the Eternal Return (1949), it names the pattern by which traditional societies are said to abolish ordinary time by ritually re-enacting the events of the beginning. On this account ritual repeats what was done in illo tempore, in the primordial time, and by repeating participates in it, so that a new year festival is not a commemoration but an actual renewal of the world. Eliade contrasted this orientation with the linear, irreversible history of the Abrahamic traditions, and treated the latter as a decisive and burdensome innovation, the point at which human beings became answerable to history rather than able to escape it. The thesis has been enormously influential and heavily criticised: for the sharpness of the cyclical-linear contrast, for treating widely differing societies as a single "archaic man", and for the evaluative preferences running through an argument presented as description.
In philosophy, eternal recurrence is Nietzsche's thought, presented in The Gay Science and Thus Spoke Zarathustra: that everything that has happened will recur, in the same order, without end. It is most often read as a test rather than a cosmological claim, the question of whether one could will one's life to repeat exactly, down to every pain, offered as the measure of an affirmative relation to existence. Some interpreters take it cosmologically, and Nietzsche's notebooks contain attempts at an argument from finite energy in infinite time.
Cyclical cosmologies themselves are widely attested and are not the same as either. Indian traditions describe vast recurring cycles, the four yugas, the kalpa as a single day of Brahma, and in Jain cosmology a wheel turning endlessly through ascending and descending half-cycles. Stoicism held to ekpyrosis, a periodic conflagration followed by an identical reconstitution of the cosmos. Mesoamerican calendrics organised time in interlocking cycles with direct cosmological consequence.
The distinction that actually matters is between traditions in which time has no direction and nothing is finally decided, and traditions in which cycles occur but liberation, judgement or renovation gives the whole a direction after all. Indian traditions are of the second kind, since moksha is an exit from the cycle rather than a position within it, which is why the familiar contrast between "cyclical East" and "linear West" is a considerable oversimplification of both.