Greetings,
The Trinity is a mystery of the faith. You get hints of the Trinity in the Old Testament. For example in Genesis, you see God’s spirit over the waters. There is talk of God as Father, and there is talk of the Wisdom of God.
In the New Testament, the nature of God is revealed in its fullness. The Bible is God’s self-revelation in the incarnation. God is Father, Son (the Word, or Wisdom of God), and Holy Spirit. They are three distinct persons, but they have one nature.
How can this be? Well, one big hint is that human beings were created in the image and likeness of God. This has to be our starting point to understand the Trinity, because no other created being was made in His own image.
But here is where the mystery really kicks in. Human beings are finite. We have a body that came from our parents, that had a beginning, and grows old and dies. God is Eternal, he is Infinite, and so if God had a body, it would fill everything. We would be somehow living in, or a part of this massive blob god, or jelly God. If you believe that, you are a pantheist. But as Christians, we believe that God is the creator of heaven and earth. He is not heaven and earth.
So for the sake of continuing this argument, let’s cross out the idea that God has a body and senses, like us. (Also setting aside the incarnation for the moment.)
What is left? Well, human beings have something that animals don’t have. We have a memory, and an intellect, and we have a will.
Our memory is kind of a repository of all that we have been. Our intellect is our active thinking about things. We can think about the past, the present, and the future. And we can also will things, and do things, like write an email, build a church, help our neighbor.
Memory, intellect, and will are not corporeal things. They are spiritual. (Some people will argue that all we are is contained in our brain, but that is another controversy we don’t need to get into right now.) Memory, intellect, and will also sum up who we are as embodied persons. One person, has these three faculties, or operations.
Now apply these three things to God. God is infinite and eternal, therefore His memory contains past, present, future all at once. God must know himself. He must be self-aware, if he is like us, therefore he must have an intellect, and that intellect must be eternal. And if he also knows himself, he must also have a will. This will is God’s love.
Again, this analogy can only take us so far, because we are finite and God is infinite. But if God created us in our image, then God the Father could be like our memory, God the Son, could be like our intellect, because he is the word of God, the self-expression of God, the self-revelation of God. And God the Father and God the Son love one another. They are a communion of persons. And that love is so infinite and eternal that it becomes the third person of the Trinity: the Holy Spirit. And that is how we have one God (nature) and three persons (operations).
All of this is speculation based on the revelation of scripture and the teaching of the church. It is a mystery that cannot be approached with science. It is only hinted at by nature, and most clearly in human nature.
Hope this helps.
God bless.