Episodes: 4.01 Lazarus Rising, 4.17 It's a Terrible Life, 5.13 The Song Remains the Same, 5.18 Point of No Return, 5.21 Two Minutes to Midnight, 5.22 Swan Song
A Vessel is a human that an angel possesses in order to manifest on Earth. The term Vessel was first used by Castiel in episode 4.01 Lazarus Rising.
Castiel: That was my mistake. Certain people, special people, can perceive my true visage. I thought you would be one of them. I was wrong.
Dean: And what visage are you in now, huh? What, holy tax accountant?
Castiel: This? This is... a vessel.
– Castiel and Dean, 4.01 Lazarus Rising
The suitability to be a Vessel is innate. Castiel makes reference when possessing Claire Novak that her suitability as a vessel is "in her blood".
Michael, when he is using the young John Winchester as his vessel, tells Dean: “You’re my true Vessel, but not my only one. It’s a bloodline…stretching back to Cain and Abel. It’s in your blood, your father’s blood, your family's blood.”
While Demons can possess any human against their will in order to manifest, Angels must contain the consent of the human before they can enter them. However, as Zachariah's treatment of Dean in attempt to get him to agree to be Michael's vessel demonstrates, this consent can be coerced.
It is unknown as to whether or not Gabriel needs a Vessel - Gabriel could have created his body. Gabriel seems to have an angelic form that's not as intense as Micheal and Lucifer's, being as how he is the youngest of the Archangels, this may be the reason he didn't seem to have difficulty in acquiring his vessel.
Zachariah's Vessel was never mentioned in Supernatural. He refers to humans as "smelly things". When Dean and Sam see Zachariah in Heaven they still see Zachariah as if he were still using his Vessel. According to Zachariah the reason they perceive him the same way in Heaven is because they're limited.
Dean: Oh, great. That's all I need is another one of you guys.
Zachariah: I'm hardly another one, Dean. I'm Castiel's superior. Believe me, I had no interest in popping down here into one of these smelly things. *Zachariah indicates his body.*
– Zachariah and Dean, 4.17 It's A Terrible Life
Anna never used a human as a Vessel. This is mainly because when an angel chooses to fall and become human, the human body they use is the most suitable "vessel" for their angelic essence. If Anna chose not to have fallen, then her "human body" would have been her Vessel. She was then raised by Mister and Misses Milton. Anna Milton was afraid of her real father - her real father is God.
Anyone within the Novak Bloodline can be used as a Vessel for normal Angels, not including Archangels.
Many Angels such as Zachariah and Castiel have told Dean that he is special and that he was chosen to be the vessel for Michael.
Gabriel reveled to Dean and Sam that they were destined to be Vessels.
PLAN B VESSELS
Plan B vessels are vessels that angels use for short periods of time. The background info on this particular type of vessel is as of now unknown.
Plan B Vessels
OTHER USES OF HUMANS BY ANGELS
Lucifer used a dead nun as a mouth piece to communicate with Azazel at St. Mary's Convent.
Prophets are also a method through which Heaven communicates with humanity, but they are not physically possessed by angels. Chuck Shurley is a prophet.