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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 Vessel represents a ship, or a piece of debris that was once part of a ship. You can access the vessel that your Part is part of with this.vessel. Particularly useful members are vessel.orbit, which gives your orbit, and vessel.mainBody, which gives CelestialBody whose influence you are currently under.