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"I don't know that a priest is required to join a particular order."

There are two main categories of priests--diocesan and order priest. Basically, diocesan are organized structurally by region in dioceses ( a geographical area). Most folks associate "priests" with the diocesan priests.
But there are also "orders" of priest's (such as the Benedictines) that are formed into groups based on a non-geographical border. They group more along a "type of spirituality (charism). (Benedictines are associated with great teaching and give the Jesuits a run for their money.)
These orders have a "command structure" apart from the diocesan system. They don't answer directly to a Bishop, but to their order's Superior who answers to the Vatican directly. The current pope, I'm quite sure has always been a diocesan priest. (They can switch with permission.)
An interesting aside is that their were no diocesan priest named "Saint" until St. John Vianney, the Cure de Ars (France).