Today we celebrate in a special way the unique Apostolic ministry bestowed by Jesus on Simon, to whom He gave the name "Peter."
"You are Peter, and upon this rock I will build my church, and the gates of the netherworld shall not prevail against it" (Matthew 16:18).
From ancient times, the bishop's church in every local Catholic Christian community has been called "the Cathedral," not because it's especially large or fancy, but because it has the bishop's "chair" ("cathedra") from which he presides at the liturgy and which representes his ministry and the responsibilities of his office. Like many of the terms and archetectural features of the early Church, it was adapted from styles and practices of imperial Rome in late antiquity (such as the "chair" where officials heard and judged cases, which had practical as well as symbolic value).
What is essential is the bishop's vocation to the service of authority passed down to them from the Apostles and established by Jesus so that His followers would be united in faith and charity in every place and time. Of course, the Bishop of Rome (i.e. "the Pope") has the central responsibility for the whole Church as the successor of Saint Peter (who was the first bishop of the originally small and beleaguered Christian community in Rome and who was martyred on Vatican hill in the year 64).
[The actual, physical chair(s) have a long history. Pictured here is ancient relic in a byzantine/early medieval setting, kept within Bernini's famous reliquary in Saint Peter's Basilica in Rome.]