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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 warship is a naval ship that is built and primarily intended for naval warfare. Usually they belong to the armed forces of a state. As well as being armed, warships are designed to withstand damage and are usually faster and more maneuverable than merchant ships. Unlike a merchant ship, which carries cargo, a warship typically carries only weapons, ammunition and supplies for its crew.