You make an offer when you present your willingness to do something. People selling a house wait for the best offer.
One person offers something, and another person accepts or rejects it. It's polite to offer guests something to drink and to offer a friend some space under your umbrella. A marriage proposal is a type of offer. A threat of violence is "an offer you can't refuse" in The Godfather. This word can mean a religious act as well, or — more grossly — the kind of offering a cat makes when it offers a dead mouse to its owner.