"Binding and loosing" is a phrase which comes from the rabbis. It refers to the authority to make decisions binding on the people of God.
This authority includes interpreting and applying the Word of God and admitting people to and excommunicating them from the community of faith. For the Jews this meant the community of Israel. For Christians this means the Church.
In Matthew 16:19 Jesus gives this authority over his Church to Peter: "Whatever you bind on Earth shall be bound in heaven; and whatever you loose on Earth shall be loosed in heaven."
In Matthew 18:18, he gives the power to all the apostles: "Amen, I say to you, whatever you bind on Earth shall be bound in heaven, and whatever you loose on Earth shall be loosed in heaven."
This singling out of Peter to bestow on him an authority which is later to be given to all the apostles shows Peter's preeminence within the apostolic college. What the apostles as a whole possessed as leaders of the Church, Peter possessed as an individual.
Of course, he, as the earthly head of the Church, also possessed powers which all the other apostles, even collectively, didn't possess: "I will give you the keys of the kingdom of heaven" (Mt 16:19).