One who is simply a brother -- a member of a religious order like the Franciscans, Dominicans, etc. -- takes vows of poverty, chastity, and obedience, but has not been ordained a priest and cannot celebrate Mass or confer the sacraments.
A priest can be either a member of a religious order (and take the same vows), or can be a diocesan priest, reporting to a bishop (in which case he takes vows of chastity and obedience).
A priest in a religious order is also a brother (yes -- a father and a brother at the same time!). Priests and brothers who belong to religious orders live in "community," that is, in friaries, monasteries, or abbeys.
What is the criteria for being either? (I assume you mean why would you be a brother who's not a priest). It depends on which one the Lord is calling you to. A lot of discernment with spiritual directors is necessary to distinguish between the two, in some cases.
Recently, I attended a conference on the subject of ethics and stem cell research, the priest was a college profesor at Georgetown and a Jesuit, an order of brothers, who had 2 doctorates--microbiology and ethics.
He thought it was fine that Bill Clinton spoke there (at Georgetown) on November 7, since he was a former president and alumnus of the school and that as a university/educational institution you have to show both sides of an issue and let the students decide which is right. I say bull....
Clinton was a slime ball, who approved partial-birth abortions, he should never have been allowed to speak at Georgetown and if this was an era before Vatican II, I don't think any university would have hosted such a person.