ousia (essence/being/substance): Important in the Trinitarian doctrine codified at the council of Nicea (325), which declared the Son to be of the same essence (homoousious) as the Father. It becomes important in Christological debates as various theologians sought to understand how the humanity of Jesus and our human nature could be homoousious (usually translated "co-essential").

hypostasis (entity, substance, "person"): Used at Nicea as a synonym for ousia (the word literally means "to stand under" i.e. "sub-stance"). Over the course of the 4th century it comes to be distinguished from ousia and taken as meaning "entity" or "individual reality" or even "distinct manner of existing." Thus in speaking of the Trinity, theologians will speak of there being one ousia (essence/being/substance) and three hypostases (entities or "persons"). In Christology there is a concern that while Christ is both fully human and fully divine, this duality of nature not be thought of as compromising the unity of Jesus Christ as a single hypostasis i.e. Jesus is only one "thing."

physis (nature): Sometimes used in Trinitarian theology as a synonym for ousia, the term really comes into its own in Christological reflection as a way of speaking of the humanity and divinity of Christ. Thus the unity of Christ's hypostasis (person) does not imply any mixing of the divine nature and the human nature.

prosopon (person, "mask/face"): This term literally means "face" and implies the "person" that we present to others. This term was used in both Trinitarian and Christological reflection as virtually synonymous with hypostasis. However, it did not convey as strong a sense as hypostasis of an actually existing entity and thus was suspected by some of indicating only the appearance of a distinct and unified act of existence.

Trinitarian theology, the "classic" formulation of doctrine is that
God is
one essence (ousia)
existing as
three persons (hypostases): Father, Son and Spirit.

Christology, the "classic" formulation of doctrine is that
Christ is
one person (hypostasis)
who possesses
two natures (physes): divine and human.

hypostatic union: A term used to describe the nature of divine and human in Christ. Christ is one because there is in his person (hypostasis) a union of the divine and human natures. The natures remain distinct because the union is on the level of Christ's hypostasis, not his natures (physes).

communicatio idiomatum (exchange of properties): A term associated with Alexandrian Christology that describes how because of Christ's unity of person, the attributes of the divine and human natures can be predicated of each other. Thus we can speak of God being born of Mary or dying on the cross.

theotokos ("God-bearer"): The usual English translation is "Mother of God." A title for the virgin Mary that was rejected by Nestorius. He argued that Mary was the mother of Christ's human nature only, and thus was not properly speaking the Mother of God. This title was accepted by Cyril of Alexandria and others on the basis of the communicatio idiomatum.