Can we explain the Trinity? On the surface, perhaps, the traditional Christian belief in a "Triune God" is non-sensical. By what possible logic does one equal three? Though Christians are the first to admit that this central belief is a great Mystery, we also believe that there are ways to explain the building blocks of the belief without claiming to know everything about it.
The Incarnation of Christ Jesus
The first issue in explaining the Trinity belongs to another reality, that of the incarnation of the Word in Jesus Christ of Nazareth. From very early on (see chapter two of Paul's letter to the assembly at Philippi in the Christian Bible), there was a distinct sense that this amazing man, Jesus, was not only a good man that God raised from the dead, but also the perfect presence of the Creator God in, through, and as a human being.
But just a second--the earliest Christians were Jewish monotheists. Every day pious Jews would say the Shemah, one of the most ancient prayers of the tradition: "Hear, O Israel: Yahweh is God, Yahweh alone." Though Jewish tradition was painfully aware of other gods in the Ancient Near East, Jews nevertheless affirmed that the God of Abraham, Isaac, and Jacob was the Creator, and as such was the only God worthy of worship--all the other gods were not gods at all, but creatures who did not deserve the worship of humankind. Early Christians absolutely believed this, but they were convinced by Jesus' resurrection from the dead and his own words and actions that he was not "just another creature," but Godself come down to rescue the world from the grip of evil powers.
Christian authors like Paul believed that Jesus did not exploit his equality with God, and along with Matthew affirmed indirectly the God the Father, Jesus Christ, and the Holy Spirit were all worthy of the worship due the one true God. The author of John reached back to the Hebrew Bible's portrayals of Wisdom as an emanation of God's character, similar to the Greek concept of the Logos or the Reason that ordered the universe. At the very beginning of his Gospel, in words designed to echo God's creation of the world in Genesis, he wrote: "In the beginning was the Word, and the Word was with God, and the Word was God." A few verses later, he made the astonishing claim, "The Word became flesh and dwelt among us; we have seen his glory, as of a Father's only son, full of grace and truth." Throughout John's Gospel, Jesus makes a series of "I AM" sayings; the author clearly wants his readers to identify Jesus as the human presence of the God of the Hebrew covenant. Though John represents the highest--the most explicit--thinking about Christ as God, there are several other clues scattered throughout all of the Gospels, Acts, and the rest of the New Testament.
Preaching in a way understood by the people
As Christianity moved from a primarly Jewish to primarily Hellenistic Pagan context, Christians found the need to accurately portray Jesus in a culture that would easily recognize people as divine (for example, the Roman Emperors were considered the human representatives of the gods) but would feel utterly scandalized by the claim that Jesus was bodily raised from the dead, since the body was a corrupt prison for the pure soul.
Thanks to the cultural shift away from Judaism, certain concepts became difficult to understand--for example, the New Testament's language of Jesus as the "firstborn of creation," which seems to imply that the Word only a creature that God used to create the cosmos, and not the fullness of the Deity itself. In ancient Jewish culture, however, the "firstborn" was the one who received all the property and authority of the tribal father, not necessarily biological son of the father. The early Church used the metaphor not in the sense of "born or created" but in the sense of "having preeminence and complete authority equal to the Father."
After many decades of struggle, Christians eventually accepted the Definition of Chalcedon as the best way to speak, without sliding into error, about both the full deity and the full humanity of Christ. Christ is, in Christian orthodoxy, one Person of the Godhead in two natures.
Knowing the Spirit as God
Intertwined with the debate about the Deity of Jesus was a debate about whether the Holy Spirit was also fully God. There was biblical support for this, too--not only the three-part formulae in the New Testament, but also language in John that calls the Spirit the teacher (chapter 15), language in Paul (1 Corinthians 12-14) that talks about the Holy Spirit giving gifts to the Church as it pleases (remember that "it" is not offensive, since God, properly speaking, does not have a gender-sex as humans use the term), and language in Acts that makes lying to the Holy Spirit the same as lying to God (Acts 5). Again, the problem became explaining the Christians' experience of Holy Spirit as God in ways that were faithful to the Gospel of Christ and yet still at least somewhat intelligible to the surrounding Greco-Roman culture. The Nicene Creed, edited for clarity at the Council of Constantinople, says about the Holy Spirit: "We believe in the Holy Spirit, the Lord, the giver of life, who proceeds from the Father"--proceeding was concept used to distinguish between Jesus, who is eternally begotten of the Father, and the Holy Spirit, who is never described as God's child in the Christian Bible. "With the Father and the Son he is worshiped and glorified; he has spoken through the prophets."
Invitation into the Mystery of God
In struggling with how to maintain monotheism whilst remaining true to what we experience, Christians therefore claim something like the following: the Father, Son and Holy Spirit are all the One Creator, and God has always been this threeness and oneness simultaneously and without contradiction or confusion. Doing the math on our end may hurt our brains, but for Christians it comes down to this: whenever we encounter Father, Son, and/or Holy Spirit, we are always experiencing all of God working together to liberate the world, because despite the necessary distinctions between them, all of the Persons shall the same One-Godness. Whether you prefer the language of philosophy or the language of the Bible to explain the Trinity, Christians have always wanted people to understand that they worship one God who reveals Godself in ways that tell us something of who God really is. When Christians try to explain the Trinity, then, this is not simply a series of mathematical gymnastic routines, but an attempt to invite people into fellowship with this amazing, knowable, but utterly strange and confounding One God.