an offering: something carried to God and presented to him as a gift, or the act of presenting it
This noun comes straight from a verb meaning to carry something to someone and set it before them (προσφέρω). So it names, very simply, what a worshiper brings to God, and the bringing of it. In the temple that could be an animal, a measure of grain, or the gift required at the close of a vow. The word itself does not say which; it says only that something was carried forward and handed over.
Because of that it points in two directions at once. Sometimes it is the gift, the thing lying there once it has been given, as when Paul pays for the offerings of four men in Jerusalem, or when Christ hands himself over as a gift to God. Sometimes it is the presenting itself, an action you can count, which is how the letter to the Hebrews uses it: the priests offer again and again, Christ offered once, and after that one offering nothing further is needed.
Its regular partner is the word for a slain sacrifice (θυσία). Where the two stand side by side, as in the psalm quoted twice in Hebrews 10, the pair sweeps in everything brought to the altar: the animal killed, and the bloodless gift set out beside it. English keeps the pairing in "sacrifice and offering," though it can leave the impression of two words for one thing.
Two verses stretch the word past the temple courts. Paul calls the non-Jewish believers he has won an offering that he presents to God, and Ephesians calls Christ's self-giving an offering with a pleasing smell, borrowing the language of the altar for a death that happened nowhere near one.