Read: Acts 3:1-10
But Peter said, “I have no silver and gold, but what I do have I give to you. In the name of Jesus Christ of Nazareth, rise up and walk!” And he took him by the right hand and raised him up, and immediately his feet and ankles were made strong. (Acts 3:6, 7 ESV)
Three little words. "In Jesus' name." Where I'm from they are almost the token ending to every prayer I have ever heard, and quite a few that I have prayed. But there is power in Jesus' name. It is a special name.
Peter knew there was authority in the name of Jesus. He had experiential knowledge of the power of Jesus. Also, he had faith—and was emboldened by the Holy Spirit—to proclaim healing for the crippled man at the gate.
The name of Jesus gets thrown around a lot by people who assume that its a magic set of syllables. Too often people wrongly assume that they can declare something supernatural will happen, sprinkle the name of Jesus around—and then POOF, that thing has to happen. That's not how it works. It has never worked like that.
Peter rightly understood that authority rested with Jesus. The power was from the Holy Spirit. And that the combined faith of the parties involved activated the work that God was looking to do in the life of the crippled man. Peter knew that in Jesus' name he had access to God, he had access to power, but he did not have control.