What is the doctrine of the Trinity?

The doctrine of the Trinity is the Christian understanding of God, which has been developed over the centuries. It was first explored in the early days of the Church by theologians such as Augustine and Gregory of Nazianzus and marks one of the key theological differences between the Roman Catholic (Western) and Eastern Orthodox Churches. However, the doctrine of the Trinity is not just an important part of Church history, it remains an important part of contemporary philosophical theology as interest in the doctrine has resurged in the modern era with the help of neo-orthodox theologians like Karl Barth.
What has continued to baffle theologians over the years about the doctrine of the Trinity is that it is based around two seemingly contradicting claims. It claims that there is one God (as Christianity is a monotheistic religion) yet God is also 'triune' as the Father, Son and Holy Spirit. This belief that God is one yet also threefold is often thought of as paradoxical and has been developed in accordance with the descriptions of God in the Old and New Testaments, where God is often presented as existing in self-differentiated 'forms' of Himself. Some point to how God appears in a different 'form' of Himself in Old Testament theophanies (such as when he appears to Moses as a burning bush in Exodus 3) but more commonly to the way that Jesus is presented as the Son of God and as being 'at one with God' in the New Testament. Examples of this include statements in the gospels (John 14:11: "I am in the Father and the Father is in me") as well as in the works of St Paul (2 Cor. 13:13: "The grace of the Lord Jesus Christ, the love of God, and the communion of the Holy Spirit be with all of you"). As a result, the doctrine of the Trinity is an attempt to defend Christianity's claim to monotheism whilst staying true to the way that God is described in Christian scripture.