Eternalism and presentism are rival positions on the ontology of time, and their religious interest lies in what they imply about divine knowledge, providence and survival.
Presentism holds that only the present exists. The past has ceased to exist and the future does not yet exist; what is real is what is happening now. It matches ordinary intuition and the felt reality of passage and change, and it makes the openness of the future straightforward, there is nothing there yet to be settled.
Eternalism holds that all times are equally real, the "block universe", and that past, present and future differ only in their relation to an observer, much as "here" and "there" differ without either being less real. Its principal support is physical. The relativity of simultaneity in special relativity means that whether two spatially separated events are simultaneous depends on the observer's frame of reference, so there is no unique universe-wide "now" of the kind presentism appears to require. Presentists have developed replies, but the argument from relativity is the main reason eternalism is the more common position among philosophers of physics.
The theological consequences are substantial. Divine eternity is classically understood in Boethius' terms as "the complete possession all at once of illimitable life", God existing outside time and apprehending all events in a single present, which dissolves the foreknowledge problem by denying that God's knowledge is fore-knowledge at all. That account sits naturally with eternalism. Open theism, by contrast, holds that the future is genuinely unsettled and that God does not know future free choices, which requires something much closer to presentism and a God who is temporal rather than timeless.
Eternalism bears on other doctrines too, and recent philosophical theology has explored them: if all times are equally real then a life is not annihilated by ending, which some theologians have used in accounts of resurrection; and the permanence of past suffering in a block universe has been used against the view as readily as for it.
Indian philosophy ran a strikingly parallel dispute. The Buddhist Sarvastivada school, whose name means "the doctrine that all exists", held that dharmas exist in past, present and future in some sense, a form of eternalism advanced partly to explain how a past act can produce a present result. Other Buddhist schools rejected it precisely because it appeared to compromise impermanence, and the resulting argument over momentariness and over whether something that no longer exists can be a cause is among the most technical in Abhidharma literature.