What is the vow that must be devoted solely to God? And does it include a person saying to another living person: "If such and such happens, I will pay you so and so?"

A vow (nadhr) is an act of worship, and directing it to other than Allah is polytheism (shirk). It is forbidden to make vows for the living or the dead to gain benefit or avert harm, because that is shirk. As for a person's statement, "I will pay you such-and-such if such-and-such happens," this is a promise for which fulfillment is sought, but it is not binding unless the promisee incurs trouble because of it.