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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.
A boat is a watercraft of modest size designed to float or plane, to provide passage across water. Usually this water is inland (lakes) or in protected coastal areas. In naval terms, a boat is something small enough to be carried aboard another vessel (a ship). A ship is a large vessel that floats on water.