I had already written an answer to the question "What is a person?" on my Trinity group on another Christian forum. I'm copying-and-pasting my answer there with some additions.
The English word "person" is the equivalent of the Greek word πρόσωπον (prosōpon) and the Latin word persona.
Boethius defines the Greek word πρόσωπον and its Latin equivalent persona as "rationabile individuum," or "a rational individual."
In the doctrine of the Trinity, we are accustomed to seeing the Greek word ὑπόστασις (hypostasis) rather than πρόσωπον.
However, these two words are synonyms and may be used interchangeably. Boethius writes,
Therefore, the Greeks called individual substances (substantias) «ὑπόστασις» since they substand (subsunt) the rest, and they are as if suppositums (suppositæ) and subjects (subjectæ) to certain accidents. Αnd for that reason we also call those substances (substantias), which they [call] «ὑποστάσεις», as if [they are] suppositums (suppositæ). Since they may also call those same substances (substantias), «πρόσωπα», we can likewise call them "persons" (personas).
A "person" is a rational individual. The English word "rational" means "having or exercising reason." Therefore, a "person" is an individual who has or exercises reason. In Greek, the English word "reason" would be translated as λόγος (logos).
Boethius further defines what may be referred to as a "person." He writes,
Now from all the definitions we have given it is clear that "person" (personam) cannot be affirmed of bodies which have no life (for no one ever said that a stone had a person), nor yet of living things which lack sense (for neither is there any person of a tree), nor finally of that which is bereft of mind and reason (ratione) (for there is no
person of a horse or ox or any other of the animals which dumb and unreasoning live a life of sense alone), but we say there is a person of a man, of God, of an angel.
In conclusion, Boethius states that only a man, an angel, or God can be referred to as a "person" because only these possess "reason."
The doctrine of "divine simplicity" says that God is without parts. The doctrine of the Trinity does not contradict the doctrine of divine simplicity. Consider fire. "Fire" is what one would consider an ousia.
οὐσία λέγεται τά τε ἁπλᾶ σώματα, οἷον γῆ καὶ πῦρ καὶ ὕδωρ καὶ ὅσα τοιαῦτα, καὶ ὅλως
σώματα καὶ τὰ ἐκ τούτων συνεστῶτα ζῷά τε καὶ δαιμόνια καὶ τὰ μόρια τούτων: ἅπαντα δὲ
ταῦτα λέγεται οὐσία ὅτι οὐ καθ᾽ ὑποκειμένου λέγεται ἀλλὰ κατὰ τούτων τὰ ἄλλα.
which is translated by W. D. Ross as,
We call 'substance' the simple bodies, i.e. earth and fire and water and everything of the sort, and in general bodies and the things composed of them, both animals and divine beings, and the parts of these. All these are called substance because they are not predicated of a subject but everything else is predicated of them.
Now, imagine a fire burning in a particular spot. Imagine now, that a man takes a stick and places it above the fire in order to spread the fire. So, holding it above the fire, the stick now catches on fire. The original fire did not diminish in intensity. It simply spread to another source.
Justin Martyr wrote,
...since he was begotten of the Father by an act of will; just as we see happening among ourselves: for when we give out some word, we beget the word; yet not by abscission, so as to lessen the word [which remains] in us, when we give it out: and just as we see also happening in the case of a fire, which is not lessened when it has kindled [another], but remains the same; and that which has been kindled by it likewise appears to exist by itself, not diminishing that from which it was kindled.
θεός, what we translate as "God," is, like the word "man," both a reference to "substance" and "person."
Οὐσία δέ ἐστιν ἡ κυριώτατά τε καὶ πρώτως καὶ μάλιστα λεγομένη, ἣ μήτε καθ' ὑποκειμένου
τινὸς λέγεται μήτε ἐν ὑποκειμένῳ τινί ἐστιν, οἷον ὁ τὶς ἄνθρωπος ἢ ὁ τὶς ἵππος. δεύτεραι δὲ οὐσίαι λέγονται, ἐν οἷς εἴδεσιν αἱ πρώτως οὐσίαι λεγόμεναι ὑπάρχουσιν, ταῦτά τε καὶ τὰ τῶν εἰδῶν τούτων γένη οἷον ὁ τὶς ἄνθρωπος ἐν εἴδει μὲν ὑπάρχει τῷ ἀνθρώπῳ, γένος δὲ τοῦ
εἴδους ἐστὶ τὸ ζῷον δεύτεραι οὖν αὗται λέγονται οὐσίαι, οἷον ὅ τε ἄνθρωπος καὶ τὸ
which W. D. Ross translates as,
Substance (οὐσία), in the truest and primary and most definite sense of the word, is that which is neither predicable of a subject nor present in a subject; for instance, the individual man or horse. But in a secondary sense those things are called substances
within which, as species, the primary substances are included; also those which, as genera, include the species. For instance, the individual man is included in the species 'man', and the genus to which the species belongs is 'animal'; these, therefore that is to say, the species 'man' (ἀνθρώπῳ) and the genus 'animal' (ζῷον), are termed secondary substances.
John of Damascus wrote,
Now we have often said already that essence is one thing and subsistence another, and that essence signifies the common and general form of subsistences of the same kind, such as "God," "man," while subsistence marks the individual, that is to say, "Father," "Son," "Holy Spirit," or "Peter," "Paul." Observe, then, that the names, divinity and humanity, denote essences or natures: while the names, "God" and "man," are applied both in connection with natures, as when we say that God is incomprehensible essence, and that God is one, and with reference to subsistences, that which is more specific having the name of the more general applied to it, as when the Scripture says, Therefore God, thy God, hath anointed thee, or again, There was a certain man in the land of Uz, for it was only to Job that reference was made.
So, the Son shares the same essence "God" that the Father has. There is no dimunition of nature when the Father begat His only-begotten Son, just as the strength and quality of the fire is not diminished when another object is kindled. The essence of God remains simple and undivided. The Son does not possess 1/3 of the essence, and the Father another 1/3, but each shares the full and perfect essence of God, undiminished and uncompound.
 Boethius. Liber De Persona Et Duabus Naturis Contra Eutychen Et Nestorium, Ch. III.
 Boethius. Liber De Persona Et Duabus Naturis Contra Eutychen Et Nestorium, Ch. II.
 Aristotle. Metaphysics, 1080β.
 Justin Martyr. Dialogue with Trypho, Ch. LXI.
 Aristotle. Categories, §5, 2a.
 John of Damascus. Exact Exposition of the Orthodox Faith, Book III, Ch. IV.