The chair of Peter represents his teaching authority. When the Pope speaks ex-cathedra (from the chair) it means he is speaking authoritatively about something that is to be held by the whole Church. This goes with the keys (see Peter and The Keys).
To understand this we need to go back to Exodus 18:13 where Moses sat in judgement over the whole people. He sat from morning until evening and was becoming worn out. His father-in-law, Jethro, enquired about this of him. Moses answered his father-in-law, “Because the people come to me to inquire of God; when they have a dispute, they come to me and I decide between a man and his neighbor, and I make them know the statutes of God and his decisions.” (Ex 18:15-16). Jethro said this was unwise and advised Moses to appoint people to judge lesser cases and Moses should “teach them the statutes and the decisions, and make them know the way in which they must walk and what they must do.” (vs 20) and Moses should just judge the more difficult cases. Thus Moses delivered God’s “statutes and the decisions”, the way in which they must walk and what they must do”.
The teachers of Israel continued to teach with authority “the way in which they must walk and what they must do”. This is why Jesus said about them “The scribes and the Pharisees sit on Moses’ seat; so practice and observe whatever they tell you,” (Mt 23:2-3). Note that Jesus is not saying they sat in a literal chair of Moses. He is speaking figuratively.
To understand this we need to go back to Exodus 18:13 where Moses sat in judgement over the whole people. He sat from morning until evening and was becoming worn out. His father-in-law, Jethro, enquired about this of him. Moses answered his father-in-law, “Because the people come to me to inquire of God; when they have a dispute, they come to me and I decide between a man and his neighbor, and I make them know the statutes of God and his decisions.” (Ex 18:15-16). Jethro said this was unwise and advised Moses to appoint people to judge lesser cases and Moses should “teach them the statutes and the decisions, and make them know the way in which they must walk and what they must do.” (vs 20) and Moses should just judge the more difficult cases. Thus Moses delivered God’s “statutes and the decisions”, the way in which they must walk and what they must do”.
The teachers of Israel continued to teach with authority “the way in which they must walk and what they must do”. This is why Jesus said about them “The scribes and the Pharisees sit on Moses’ seat; so practice and observe whatever they tell you,” (Mt 23:2-3). Note that Jesus is not saying they sat in a literal chair of Moses. He is speaking figuratively.