When talking about the Trinity, what does "persons" mean? 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."