Trinitarianism and other Christian conceptions

To trinitarian Christians (which since post-apostolic times has represented the vast Christian majority), God the Father is not at all a separate god from the Son (of whom Jesus is the incarnation) and the Holy Spirit, the other members of the Christian Godhead. Trinitarian Christians describe these three persons as a Trinity. This means that they always exist as three distinct "persons" (Greek hypostases), but they are one God, each having full identity as God himself (a single "substance"), a single "divine nature" and power, and a single "divine will". Theologian Alex Nicholson alluded Trinitarianism to how water can be a liquid, a solid (ice) and a gas, but maintain the same elemental properties.

Similar to the way that the Tritarian God is the Father, the Son, and the Holy Spirit (different in context) and essentially the same being.
Other Christians, nonetheless, held alternative ideas about the Trinity. A handful have described the Father, Son and Spirit as each a distinct, eternally existent being (tritheism), or as a different "manifestation" of a single being (modalism). Some have theorized that the relationship of Father and Son began at some point probably outside of normal "history" (Arianism); and others have believed that God became a Father when he uttered his creating Λογος ("logos" or "word"), who is both a principle of order and a living being to whom God bears the relationship as Father (some gnostics). Others found strong affinity with traditional pagan ideas of a savior or hero who is begotten by deity, an idea of the Father similar to Mithraism or the cult of the Roman emperor.

For many Christians, the person of God the Father is the ultimate, and on occasion the exclusive addressee of prayer, often in the name of Jesus Christ. The Lord's Prayer, for example, begins, "Our Father who art in Heaven...."
In the New Testament, God the Father has a special role in his relationship with the person of the Son, where Jesus is believed to be his Son and his heir (Hebrews 1:2-5). According to the Nicene Creed, the Son (Jesus Christ) is "eternally begotten of the Father", indicating that their divine Father-Son relationship is not tied to an event within time or human history. See Christology. The Bible refers to Christ as the beginning of God's creation, and hence as God's "firstborn."
In Eastern Orthodox theology, God the Father is the "source" or "origin" of both the Son and the Holy Spirit; in Western theology, all three hypostases or persons have their origin in the divine nature instead. The Cappadocian Fathers used this Eastern Orthodox monarchian understanding to explain why trinitarianism is not tritheism: "God is one because the Father is one," said Basil the Great in the fourth century. In the eighth century, John of Damascus wrote at greater length about the Father's role:

Whatsoever the Son has from the Father, the Spirit also has, including His very being. And if the Father does not exist, then neither does the Son and the Spirit; and if the Father does not have something, then neither has the Son or the Spirit. Furthermore, because of the Father, that is, because the Father is, the Son and the Spirit are; and because of the Father, the Son and the Spirit have everything that they have.