What are the different models of the Trinity in the Christian tradition?

The Psychological and Social Models of the Trinity
The Bible teaches that there is only one God. At the same time, it teaches that the Father, Son, and Holy Spirit are each fully God. For this reason the church has always affirmed the doctrine of the Trinity, which teaches that God has one substance (ousia) but eternally and fully exists as three distinct Persons (hypostases in Greek, personae in Latin).

The question left unanswered is how we are to understand the relationship between the substantial unity and personal plurality within the Godhead. Throughout church history, two distinct models have been proposed.

The first model goes back to the second-century apologists and was made famous by Augustine. It is usually called the psychological model of the Trinity, for it likens the unity and diversity of the Godhead to the unity and diversity of the human self. According to Augustine, the unity of the Father, Son, and Holy Spirit is like the unity of the mind (thoughts), heart (emotions), and volition (will) of a person. A different version was put forth by Jonathan Edwards in the eighteenth century. He argued that as the human psyche consists of a self, a self-image, and a relationship between the self and the self-image, so the Godhead consists of a self (Father), a perfect self-image (Son), and a perfect loving relationship (Spirit) between the self and the self-image.

While many have found the psychological model helpful, others have objected to it on the grounds that it is not faithful to the biblical data. The Bible depicts the Father, Son, and Holy Spirit as three distinct Persons, they argue, not three distinct aspects of one Person. Hence, they have proposed that we should think of the unity of the Trinity more like the unity of the mind, heart, and will of three people. This has been labeled the social model of the Trinity.

To some defenders of the psychological model, the social model borders on tri-theism (the belief in three separate gods). To some defenders of the social model, the psychological model borders on modalism (reducing the three Persons to three modes of one person). Still others argue that since models are only models, not exact replicas of reality, we may affirm both models as valid in capturing distinct perspectives on a God whose full reality defies exhaustive description.