I will explain the "keys of the kingdom of heaven", and binding and loosing on earth and in heaven.
The keys both bind and loose.
Peter first binds the Church to the Law at the beginning of Acts. For 15 chapters the Church is run as a sect of Judaism out of the temple in Jerusalem based on keeping the Old Covenant Law. If the disciples had not kept the Law they would have been stoned to death at the orders of the leaders of the Jews. God abided by this decision, blessing the Church in many ways - what Peter bound on earth was bound in heaven.
The Law binds:
Ga.3.23 Before this faith came, we were held prisoners by the law, locked up until faith should be revealed.
Ga.4.9 But now that you know Godor rather are known by Godhow is it that you are turning back to those weak and miserable principles? Do you wish to be enslaved by them all over again?
Freedom from the law loosens:
Ro.8.1-2 Therefore, there is now no condemnation for those who are in Christ Jesus, 2 because through Christ Jesus the law of the Spirit of life set me free from the law of sin and death.
Ga.5.1 It is for freedom that Christ has set us free. Stand firm, then, and do not let yourselves be burdened again by a yoke of slavery.
At the council of Ac.15 it is Peter who again uses the keys of loosening to cast off the Law from the Church:
Ac.15.5-7a,10 Then some of the believers who belonged to the party of the Pharisees stood up and said, "The Gentiles must be circumcised and required to obey the law of Moses." 6 The apostles and elders met to consider this question. 7 After much discussion, Peter got up and addressed them: 10 "Now then, why do you try to test God by putting on the necks of the disciples a yoke that neither we nor our fathers have been able to bear?"
The Law was abandoned. The Church was freed from the Law, and God again honored that decision as loosed in heaven. That is the extent of the binding and loosing that Peter had with the keys given to him by Christ - the keys to bind and loose the Law on the Church.
At the council of Ac.15 it is Peter who again uses the keys of loosening to cast off the Law from the Church:
Acts 15
20 But that we write unto them, that they abstain from pollutions of idols, and from fornication, and from things strangled, and from blood.
We can see that it is a far cry from the laws the Jews had and many of the churches have today.
And her father hear her vow, and her bond wherewith she hath bound her soul, and her father shall hold his peace at her: then all her vows shall stand, and every bond wherewith she hath bound her soul shall stand. (Numbers 30:4)
Elijah bound the heavens for 3.5 years and then loosed them, and who James invokes as an example of what believers can do in James 5: 16-18)