The short answer: If you pay your actors (Equity or non-Equity), you're a professional group. If you don't pay your actors, you're an amateur group.
The long answer:
Amateur Performances include performances as staged by little theaters, community theaters and/or drama associations; colleges, universities, high school and other school groups; churches and other religious institutions; and/or puppetry theaters, clubs, and other amateur organizations or groups, (a) irrespective of the size or nature of the audience; (b) regardless of whether or not an admission fee is charged; and (c) regardless of whether the performance is given by paid or unpaid actors and production staff. Amateur Performances does not, however, include performances given by any theater or group using a cast which consists entirely, or of a majority, of paid actors and production staff and/or that has normally been known to handle stock pursuant to one of the Actors Equity Association agreements governing employment of actors.
Professional Performances include performances given in any theater or by any group using a producing cast which consists entirely, or of a majority, of paid actors and production staff, and/or has normally been known and handled as stock, including puppetry theaters, Council of Off Loop Theaters (COLT), Council of Stock Theaters (COST), League of Resident Theaters (LORT), regional theaters, dinner theaters, repertory productions, etc.