The triune God is defined as three persons equal in nature. What does "persons" mean in this statement? Can the word "persons" be replaced with "personalities" or "roles"?
The classic definition of "Person" is that given by Boethius in De persona et duabus naturis: an individual substance of a rational nature).
St. Thomas Aquinas explained the preceding definition in terms that practically constitute a new definition: a substance, complete, subsisting per se, existing apart from others (Summa Theologica, III, Q. xvi, a. 12, ad 2um).
Regarding the persons of the Blessed Trinity, for the constitution of a person it is required that a reality be subsistent and absolutely distinct. The three Divine realities are relations, each identified with the Divine Essence. A finite relation has reality only in so far as it is an accident; it has the reality of inherence. The Divine relations, however, are in the nature not by inherence but by identity. The reality they have, therefore, is not that of an accident, but that of a subsistence. They are one with ipsum esse subsistens (Subsistent Act of Existing Itself).
A much, much more detailed discussion of this topic -- with citations that could give you a few years more reading -- can be found in the Catholic Encyclopedia's entry on "Person."
Personalities, possibly, but roles no. To say that God merely exists as one person filling three jobs (roles) or having three "faces" is a heresy known as modalism. The exact formulations are always contentious, and the truth is there is no satisfactory analogy, because there is no terrestrial equivalent.
The tricky balance of the Trinity is to maintain the distinct personhood of the Three individuals, even though they are so close - closer even than the "one flesh" that man and wife are supposed to become.
The answer is a bit both. The personalities and roles could both explain the persons in the Trinity. But it is important to remember that the three persons are not separated from one another.
God cannot be constituted of different parts and cannot be subdivided. Is being in single, numerically one, exempt from any composition. The man on the other hand has different parts like the material and the non-material. But God is spirit and is not subject to division. This unity is not incompatible with the notion of the Trinity, because unity is not the same has one unit. One unit is basically one single element. God's unity allow for personal distinction in his divine nature, while still maintaining that the divine nature is numerically one and eternally one. The unity implies that the three persons in the trinity are not three essences separated inside the divine essence.
Henry C. Thiessen, Lectures in Systematic Theology, chapter 9
I only had the book in French, so the quote is a rough translation.