Peter’s life is so important to the Church that he has more than one feast day during the year. This feast is another way the Church celebrates this.
The chair of St. Peter is a symbol of authority, as Jesus refers to in Matthew 23:2, when he commands and warns the crowd and his disciples, “The Scribes and Pharisees have taken their seat on the chair of Moses”. The “chair” is metaphorically referring to the teaching and judging office that Moses held in the Name of God. The name that Jesus gives to Peter is a second metaphor for an office of teaching and judging, and to drive the point home, Jesus uses a third metaphor by speaking of the “Keys of the kingdom of Heaven” Mathew 16:19. The “power of the keys” is used in many ways; some are specific to the Office of St. Peter the Papacy, others are shared with those ordained to priestly ministry, the Sacrament of Confession. We pray for these human leaders of the Church chosen by God, so they are faithful ministers of God’s grace to us.
This feast day is an anniversary and celebration of the “authority” of the Papacy, and the particular mission entrusted to St. Peter by Jesus. The Roman Missal explains the symbol of the chair and emphasizes the mission of teacher and pastor of Christ conferred upon Peter. He formed Peter in His person and in St. Peter’s successors, a visible foundation of unity of the Church. The literal chair is a fixed seat of the supreme Pontiff and the Bishops, permanently placed in the Mother Church of the Diocese – the Cathedral. It is the symbol of the Bishop and his ordinary magisterium in the local church. The Chair of St. Peter indicates his position in the Apostolic College, established by the express will of Jesus, who assigned the task of “feeding” the flock, to lead the People of god, the Church.
Some interesting facts about “the Chair of St. Peter”:
What is the spiritual significance of this feast day?
This is a very ancient tradition, proven to have existed in Rome since the fourth century. On it we give thanks to God for the mission he entrusted to the Apostle Peter and his Successors. Celebrating the "Chair" of Peter means recognizing it as a privileged sign of the love of God, the eternal Good Shepherd, who wanted to gather his whole Church and lead her on the path of salvation.
Did St. Peter really sit in this chair?
The Chair of St. Peter, represented in the apse of the Vatican Basilica is a monumental sculpture by Bernini. It is a symbol of the special mission of Peter and his Successors to tend Christ’s flock, keeping it united in faith and in charity. Less emphasis is placed on the archaeological authenticity of the chair than on its spiritual significance.
Does the pope have to sit in the physical Chair of Peter be infallible?
No. Although the pope's infallible pronouncements are called ex cathedra (Latin, "from the chair") statements, he does not have to be sitting in the physical chair (which is rather high off the ground in any case). In fact, he doesn't have to be seated at all. He simply has to use the fullness of his authority as the successor of Peter to definitively teaching a particular matter pertaining to faith or morals. This use of the full extent of his teaching authority is referred to figuratively, as him speaking "from the chair" of St. Peter. It's a figurative expression, not a reference to the physical object.
Adapted from “9 things you need to know about the Chair of St. Peter" by Jimmy Akin on ncregister.com