The Trinity can be explained by using analogy, which is the only way it can be explained and somewhat comprehended, as many have done here. Body, mind, spirit (three components of one unity). Ice, water, steam (three states of existence of the same substance). Sister, Daughter, Mother (three ways of explaining different people experience the same being). Walk, Hit by pitch, Base Hit (three ways of getting to First Base, none of which require a Paradise by the Dashboard Light).
But no, I cannot understand it. And I used to teach it. Go figure.
The problem is not, as many have said, that a human trying to comprehend God is like an ant trying to understand the interstate system. Well, that is a part, albeit a very small part, of the problem. God, the Christian understanding of God anyway, is that God is something other than human, so different and vastly superior as to be beyond our comprehension (this leaks over into a lot of God thinking – How did the snake even get into the Garden of Eden in the first place? God placed it there. Why? And away we go with theory and theology and eventually doctrine).
The problem is the concept of Trinity just ain’t biblical. Or not explicitly biblical – the passages used to support (defend, buttress, explain, and so on) the concept seem almost anti-trinitarian when read without a bias.
What happened is this: Jesus (Emmanuel – God with us) came to be in the Jewish community where there was, and could only ever be, only one God. And Christianity grew out of this religion – more literally, was tossed out of Judaism – so there was a basic conflict (the conflict which resulted in the tossing in the first place): Jesus was God. Blaspheme to devout Jews. (Etymologically speaking, when Peter said “You are the Christ”, it was heard as “You are YHWH”.)
Because of this, some taught Jesus became God at some point (still did not avoid the one-God issue). Others said he was only God’s son (which made him somehow less than a full God, never mind the creed: Fully God and Fully Man concept, which no one really grasps either). And so on.
To solve the problem, a Church Council decided Jesus must somehow be one with God and birthed Trinitarian conceptualization of the All-Powerful Being. Co-equal and all that. But then came the Paraclete (Holy Ghost) problem. Jesus said the comforter could not come while he was on earth, therefore, the Holy Spirit must be some other emanation/experience/face/way of understanding God, now divided into God the Father, God the Son and God the Holy Ghost.
Perhaps there are other ways God can be experienced we have yet to catalog – maybe the concept should be Quintitarian (5 ways), Septatarian (7 ways) or Infinitamtariain (Infinite ways), bet we settled on three. (Like the Rabbits in Watership Down. . 1, 2, 3 and too many to deal with, perhaps we cannot handle any more.)
Maybe the best way God can be understood is Father (loving, disciplinarian, etc), Son (advocate, etc) and Friend (Holy Ghost – helper).