(This example of how to think about the Trinity is taken from oca.org.)
The Father, the Son and the Holy Spirit are called three divine persons. Person is defined here simply as the subject of existence and life—hypostasis in the traditional church language.
As the being, essence or nature of a reality answers the question “what?”, the person of a reality answers the question “which one?” or “who?” Thus, when we ask “What is God?” we answer that God is the divine, perfect, eternal, absolute . . . and when we ask “Who is God?” we answer that God is the Father, the Son, and the Holy Spirit.
The Church has explained this tri-unity of God by using such an example from worldly existence. We see three men. “What are they?” we ask. “They are human beings,” we answer. Each is man, possessing the same humanity and the same human nature defined in a certain way: created, temporal, physical, rational, etc. In what they are, the three men are one. But in who they are, they are three, each absolutely unique and distinct from the others. Each man in his own unique way is distinctly a man. One man is not the other, though each man is still human with one and the same human nature and form.
Turning to God, we may ask in the same way: “What is it?” In reply we say that it is God defined as absolute perfection: “ineffable, inconceivable, invisible, incomprehensible, ever-existing, and eternally the same.” We then ask, “Who is it?”, and we answer that it is the Trinity: Father, Son, and Holy Spirit. In who God is, there are three persons who are each absolutely unique and distinct. Each is not the other, though each is still divine with the same divine nature and form. Therefore, while being one in what they are; the Father, the Son, and the Holy Spirit are Three in who they are. And because of what and who they are—namely, uncreated, divine persons—they are undivided and perfectly united in their timeless, spaceless, sizeless, shapeless super-essential existence, as well as in their one divine life, knowledge, love, goodness, power, will, action, etc.
It is the mystery of God that there are Three who are divine; Three who live and act by one and the same divine perfection, yet each according to his own personal distinctness and uniqueness. Thus it is said that the Father, the Son and the Holy Spirit are each divine with the same divinity, yet each in his own divine way. And as the uncreated divinity has three divine subjects, so each divine action has three divine actors; there are three divine aspects to every action of God, yet the action remains one and the same.
We discover, therefore, one God the Father Almighty with His one unique Son (Image and Word) and His one Holy Spirit. There is one living God with His one perfect divine Life, who is personally the Son, with His one Spirit of Life. There is one True God with His one divine Truth, who is personally the Son, with His one Spirit of Truth. There is one wise and loving God with His one Wisdom and Love, who is personally the Son, with His one Spirit of Wisdom and Love. The examples could go on indefinitely: the one divine Father personifying every aspect of His divinity in His one divine Son, who is personally activated by His one divine Spirit. We will see the living implications of the Trinity as we survey the activity of God in his actions toward man and the world.