A monsignor is essentially an honorary bishop. The title means
My Lord, as in a courtly context. (In many countries, bishops are addressed as Monsignor.) There are actually three different levels of monsignor: Chaplain of Honor to His Holiness, a Prelate of Honor, and an Protonotary Apostolic. Most Monsignori in parishes and dioceses are in the second category, with a handful in the third. They typically have the letters P.A. after their names.
The title is honorary, and it allows the monsignor to wear a little purple trim around his cassock and biretta, though these are often not seen in everyday wear. A monsignor is still a priest, and the title does not give him any special responsibilities, as it is more a recognition of service to the Church. A bishop submits names to Rome for approval. and the Holy Father signs off on it. Some bishops do not submit names to become monsignori often or even at all, while other bishops do so more frequently. In my previous diocese, a former archbishop from the late seventies to mid nineties named no monsignors; his successor named about 25 in 16 years.
Until 1968 Monsignori could celebrate Pontifical Masses four times a year, essentially dressed as bishops. In many dioceses today several high-ranking chancery figures (chancellors and vicars general who are not bishops) tend to have this title, as do pastors of large, successful parishes, and others who have contributed to the life of the diocese in some special way. Again, it depends on the bishop’s style. The title is The Reverend Monsignor ________, and the priest is adressed as Monsignor. (Pre-1968 there were also Very Rev. Msgr. and Rt. Rev. Msgr., but these were done away with.)
A priest today who leads a deanery is a vicar forane and is titled The Very. Rev. Fr. ______ or The Very Rev. Msgr. _____, if he is also a Monsignor. He also may use the letters V.F. after his name, in lieu of the title.
In the picture below, taken at St. Patrick’s Cathedral in New York, Auxiliary Bishop Dennis Sullivan is on the left, and Msgr. Robert Ritchie, rector of the cathedral, is on the right.