Manifestation is how supernatural entities choose to appear on Earth, whether it be by assuming human form or by possessing a person, either voluntarily or involuntary.
Gods and ghosts don't need a human host to manifest, instead they assume human form, as do the Four Horsemen of the Apocalypse. According to Joshua, God's presence is somewhere on Earth, although it remains unclear as to whether or not he's in human form (see Chuck for a further discussion of this). Both angels and demons alike can dematerialize, manifesting in another time and place using the same or different host. Dragons assume human form while occupying enclosed or small environments.
Apparently, Jimmy Novak - Castiel's chosen vessel - can perceive the angel's true visage, suggesting that all vessels can perceive the visages of their chosen angel. Unlike demons who possess humans involuntarily without their consent, archangels and angels require the permission of the human host for possession to be acceptable.
Key terms pertaining to Manifestation
Possession - The term "Possession" describes the process in which a supernatural entity takes over a human's body voluntarily or involuntarily depending on the entity, effectively controlling the thoughts, speech and actions of the host.
Vessel - The term "Vessel" describes a human's body that an Angel occupies or inhabits after bearing the consent or permission given by the host, as it is required for all angels.
Meatsuit - colloquial term used by demons adn hunters alike to describe a human body when it is playing host to a demon.
Enochian Chant - The term "Enochian" refers to many things concerning angels in general. There is a specific Enochian chant that can forcibly remove Angels from their Vessel, sending them back to Heaven.
Exorcism - The term "Exorcism" describes a process Demons experience that forcibly removes them from their human host, sending them back to Hell.
Binding Link - The term "Binding Link" refers to a specific bond or connection that Demons can use to bind or lock themselves inside a human's body, in which case exorcism proves ineffective against the demon within the host.
Ectoplasm - The term "Ectoplasm" refers to a process or cycle that entities like Ghosts use to successfully materialize on Earth. Specifically, this term was used in a similar sense in the 1984 film, Ghostbusters.
The most unique element about angels (in Supernatural) that isn't in typical lore is the notion that they have to possess. They control a human host in much the same way demons do. The only difference is that angels have to ask permission and demons don't. That provided us with all sorts of morality plays to work through because angels are nominally good, but they are putting human victims through this horrific experience just because the humans were devout enough to give their will away.
– Eric Kripke, Supernatural: The Official Companion Season 4