God is both a who and a what. When we talk about what God is, we refer to his attributes, what makes God uniquely God. Thus God is all powerful, all knowing, all kind, all loving and so on. When we talk about who God is, we refer to the Trinity. Putting the what and who together we can say, there is one God who exists as three co-eternal, co-equal persons, the Father, the Son and the Holy Spirit.
Another way to explain the Trinity is: (1) The Father, Son and Holy Spirit are distinct Persons; (2) each Person is fully God; (3) there is only one God.
While we believe the Trinity is a mystery it is not illogical. God is one in one way and three in other, different ways. God is one with respect to essence (what it means to be God), purpose and unity. God is three with respect to Persons, functions and relationships.