Question: "What does it mean that the Trinity is God in three Persons?"
Answer: When we talk about the Father, the Son, and the Holy Spirit as being “Persons,” we do not mean they are human beings or that they are like mankind in any way. In our everyday language, though, that is how the word person is often used, so it is understandable that some confusion surrounds references to the three “Persons” of the Trinity.
When we talk about God, we are using the word Person to show that the Father, Son, and Holy Spirit each have personhood or personality. That is, the Father, Son, and Holy Spirit each have intellect, emotion, and volition. Any being with rationality, emotion, and a will can be considered a person; thus, human beings are persons, but so are angelic beings and the Divine Being. The definition of person cannot include physicality for the simple reason that human beings do not cease to be persons after death. A dead person’s body is left behind to decay, but his true self—his personhood—lives on in either heaven or hell.
When we speak of God existing in three Persons, we mean that God’s existence is comprised of three distinct centers of intellect, emotion, and will. Each Person of the Trinity had a unique role in creation and in the salvation of mankind. The Holy Spirit is unique and is not the Father or the Son (He proceeds from the Father and the Son, John 15:26). The Father and the Son are also unique (when Jesus prayed to the Father, He was not praying to Himself, Luke 23:34). Each is God, but each is a separate “Person.” Using the word person is one of the only ways our language has to describe this concept.
All three Persons of the Trinity comprise the one, perfectly unified God. They share the same nature and essence, and they are all the same God, but each individual Person of the Trinity is distinct and unique. The fact that God exists in three Persons is important for several reasons. For instance, God is love (1 John 4:8). But, in eternity past, before God created any other being, could He have truly been love? That is, can love exist where there is no one to be loved? Because God exists in three co-equal, co-eternal Persons, love exists, too. Eternal love has been expressed eternally among the Persons of the Godhead. The Father, Son, and Spirit have always loved each other, and so love is eternal.
Once we lay aside the notion that a “person” can only be a “human person,” we can more readily understand how God can correctly be said to exist in three “Persons.”