“Person” is the English translation of the Latin term persona and the Greek term hypostasis. “Person” was used to denominate the individual and irreducible existence of the divine Father, Son, and Spirit respectively. In fact, the most direct definition of “person” would be the rather uninspiring expressions “something that exists” or “mode of subsistence.” In Thomism it would gain the added notion of “subsistent relation,” though that definition is still very much contested. In no case, however, did it have the modern basically univocal meaning of human “person,” still less “personality.”
Richard Muller explains how the older definition of “person” differed from its contemporary meaning here. Bavinck states plainly:
The Christian church and Christian theology, it must be remembered, never used the word “personality” to describe God’s being; and in respect of the three modes of subsistence in that being, they only spoke of persons reluctantly and for lack of a better term. (Reformed Dogmatic vol. 2, pg. 50)
Person really doesn’t mean “person,” at least not in the ordinary sense we use it today. This becomes clear when you return to hypostasis. The Greek term really means “substance.” It appears in the Bible this way in Hebrews 1:3. In the older Greek formulation, the Trinity was three substances in one essence. The Latin found this confusing, as it used substance and essence interchangeably, and so persona was offered up for an equivalent term.
It is important to keep this history in mind because “person” simply isn’t a “Biblical-Theology” term at all. It is a product of systematics, history, and translation. So much of the modern “personalism” fascination is immediately derailed when this fact is admitted. The Trinity was never used to found some theory of “personal” activity over and against “natural” activity, or what have you, nor was it in any way attempting to found some sort of psychology or sociology. The terms are historical accouterments added to the discussion after dogmatic theory was long at work.
It is also very important to note that the divine “person” is unlike the human “person” in a number of ways. Divine persons are always persons. They are never people. Further, a human person is an individual in way that the Divine persons are not. The Divine persons fully share the singular divine nature, without every having “more” or “less” than one another. The Divine persons also share a singular mind, will, spirit, and energy. This cannot be said of human persons, nor of corporations or societies.
So in summary, Trinitarian theology uses “person” to refer to a distinct and irreducible mode of the divine being. There are three persons in the one God. The three are unique in their mode of existing, but they share the same existence. They are relations of the one essence.