Today we celebrate in the Church the Feast of the Chair of St. Peter. (Yeah, one of those 'you Catholics worship anything!' days)
First, it is a sign of authority and teaching. Think to the passage in the Gospel of the Unjust Judge. He sits on his judgement chair at the gates to the city. So to sit in judgement is a sign of that authority which the one who sits possesses.
It is also a sign of unity. For the community gathers around the one presides over them, listening and learning.
Therefore, we see that Peter's Chair is a sign of the authority that is contained within the one who sits upon it. But also the sign of unity throughout the Church. And those keys which have been given to Peter have been passed down through the ages to Francis today.