decay: the breaking down of a dead body in the grave
Every use of this word in the New Testament belongs to one argument, made twice. Peter in Acts 2 and Paul in Acts 13 both quote the same line from Psalm 16:10, "you will not let your Holy One see decay", and both draw the same conclusion from it. David died, was buried, and his body broke down like anyone else's, so the promise in the psalm was never finally about him.
The phrase "to see decay" is a Greek way of saying "to undergo decay". Nobody is watching anything. The word names what a grave does to a body over time. Set beside the broader Greek word for decay (φθορά), which stretches to moral rot and to the perishing of everything created, this one stays fixed on the physical outcome of death.
One verse steps outside that quoted phrase. In Acts 13:34 Jesus is raised "no more to return to decay", which treats decay less as an event than as a condition, a place a person can be in or out of. Resurrection is the way out of it.
It is worth knowing that the Hebrew of the psalm uses a word (שַׁחַת) that normally means "the Pit", the grave itself, while the Greek translation quoted in Acts names what the grave does to a body. Both point at the same rescue, but the Greek sharpens it into a claim about flesh that did not rot.