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Vessel is a catch-all term, like 'watercraft', which describes any floating object used for the carriage of people or goods. Generally smaller and less complex vessels are 'boats', whilst larger and more complex vessels are 'ships'. As a general rule, you can put a boat on a ship, but you can't put a ship on a boat.
Cecil replies: Well, we could get technical, I suppose. Among sailing vessels, the distinction between ships and boats is that a ship is a square-rigged craft with at least three masts, and a boat isn't. With regard to motorized craft, a ship is a large vessel intended for oceangoing or at least deep-water transport, and a boat is anything else. But that's too much to remember. Try this: ships have to be big enough to carry boats, and boats have to be small enough to be carried by ships. There are exceptions, of course. Many commercial fishing craft, for example, are sizable oceangoing vessels, yet they're almost invariably called boats.