A vessel is the human which an angel possesses in order to manifest on Earth. It is unclear what characteristics make a human suitable as a vessel. There is much talk by the angels in relation to Sam and Dean of destiny. Castiel makes reference when possessing Claire Novak that her suitability as a vessel is "in her blood."
While demons can possess any human against their will in order to manifest, angels must gain the consent of the human before they can enter them. However, as Zachariah's treatment of Dean in attempts to get him to agree to be Michael's vessel demonstrates, this consent can be coerced.