From The Rad Trad: The ceremonies are a magnificent display and series of prayers which blend the norms of a Pontifical Mass (Mass celebrated by a bishop) with elements which emphasize and explain the role of the Papacy. Some of these rites are very ancient, as are many of the vestments used, while some are baroque innovations which reflect the political and devotional moods of the Counter-Reformation, a defining era in Catholic history.
The Papal Coronation Office and Mass are votive liturgies, doubles of the first class. This rite may be celebrated on any day outside of the Sundays of Advent, of Lent, the first three days of Pentecost and Easter, Holy Week, Christmas Eve, and Christmas Day. This liturgy may even supersede most holy days. In fact the rubrics demand the Mass and coronation be celebrated on the nearest Sunday or Holy Day. Innocent III was crowned on February 22, the feast of St. Peter's Chair in Antioch, and Paul VI on June 30th, the commemoration of St. Paul. No commemoration is made of the replaced day.
The procession into St. Peter's is preceded by the standard servers of Mass, who, when the Pope says Mass, are members of the Papal Household, since abolished by Paul VI. Then follow the Canons of St. Peter's Basilica, then the attending bishops, followed by cardinals in choir dress, then the ministers of the Mass, and finally the Pope, carried on the sedia gestatoria, flanked by two large fans made of ostrich feathers, and surrounded by members of the various Roman honor guards for security.
The ministers of Mass are: the Latin and Greek subdeacons, the Latin and Greek deacons—all fully vested according to to their rites, the deacons of honor, the assistant priest, the chaplains, the masters of ceremony, and the celebrant—the Pope. These roles, except for the Greek ministers, would be filled according to one's place in the Roman Curia: remember, this is a Mass said by the Pope and served accordingly by his brethren. The Greek ministers would typically be conscripted from the Byzantine monastery in Rome, which exists to this day.
On the altar are two Papal tiaras to the Epistle (right) side and two episcopal mitres to the Gospel side (left). Seven candles adorn the altar, with the crucifix in the center, and another row of candles in front of the altar over the tomb of St. Peter. Relics of Saints are often placed on the altar, as are two small statuettes of Ss. Peter and Paul. The Pope and the ministers of Mass arrive at the main altar of St. Peter's Basilica, with the sedia gestatoria left right behind the Pope, as it will be needed momentarily.
The cardinals who are not ministers of the Mass or acting as chaplains to the Pope at his throne go to their choir stalls in the apse of the Basilica. Their choir dress for Papal Mass is something of a remnant of ancient concelebration mixed with the dress of Canons: They wear a red cassock with a train, a rochet, their pectoral cross, the mitre, and, depending on rank, a dalmatic, chasuble, or cope—for cardinal deacons, cardinal priests, and cardinal bishops respectively.
The coronation of the Pope can take place in any number of locations. Some say it was always at St. Peter's, while others indicate that Popes were crowned at St. John Lateran. Leo XIII and Benedict XV were crowned in private ceremonies in the Sistine Chapel. Pius VII was crowned in exile using a tiara of paper mache at San Giorgio's in Venice. In recent memory the Papal Coronation, and the new rite inauguration, has taken place in St. Peter's Square. Pius XII and John XXIII were crowned on the balcony overlooking the square, while Paul VI, who held the Mass on the steps of the Square, was crowned at the throne erected in front of the doors to the Basilica.