Acts 3:1-10: 1 One day Peter and John were going up to the temple at the time of prayer—at three in the afternoon. 2 Now a man who was lame from birth was being carried to the temple gate called Beautiful, where he was put every day to beg from those going into the temple courts. 3 When he saw Peter and John about to enter, he asked them for money. 4 Peter looked straight at him, as did John. Then Peter said, “Look at us!” 5 So the man gave them his attention, expecting to get something from them. 6 Then Peter said, “Silver or gold I do not have, but what I do have I give you. In the name of Jesus Christ of Nazareth, walk.” 7 Taking him by the right hand, he helped him up, and instantly the man’s feet and ankles became strong. 8 He jumped to his feet and began to walk. Then he went with them into the temple courts, walking and jumping, and praising God. 9 When all the people saw him walking and praising God, 10 they recognized him as the same man who used to sit begging at the temple gate called Beautiful, and they were filled with wonder and amazement at what had happened to him.
"We immediately see the disciples using Jesus’ name as an indication of the authority under whom their miracles would continue to manifest. It’s important to remember that the miracle was granted by God, through faith the disciples developed from witnessing Jesus’ miraculous works on Earth. The disciples’ use of Jesus’ name outwardly acknowledges the source of their power rather than trying to take credit for the miracle. “In Jesus’ name” is not a magical phrase or meaningless incantation that we use to get what we want. We use Jesus’ name in prayer to glorify him, to acknowledge that the power of God was manifested through Jesus, and that we appreciate and acknowledge Jesus’ sacrifice for us. God used Jesus to deliver a new message to us, and through use of his name, we continue God’s plan.." - Bible Blender
Acts 3:19: Repent, then, and turn to God, so that your sins may be wiped out, that times of refreshing may come from the Lord.
Thomas Watson
“The Nature of True Repentance, Part 1 I shall next show what Gospel repentance is. Repentance is a grace of God’s Spirit whereby a sinner is inwardly humbled and visibly reformed. For a further amplification, know that repentance is a spiritual medicine made up of six special ingredients: 1. Sight of sin 2. Sorrow for sin 3. Confession of sin 4. Shame for sin 5. Hatred for sin 6. Turning from sin If any one is left out, it loses its virtue.” ― Thomas Watson, The Doctrine of Repentance
Acts 3:21: Whom the heaven must receive until the times of restitution of all things, which God hath spoken by the mouth of all his holy prophets since the world began.
The end result of Jesus' work is the regeneration of all things (Matt 19:28), and the restoration of all things; (translated as "restitution" in the A.V. and as "restoration" in the R.V.). In this verse, the word apokatastasis in the papyri means "the restoration of estates to rightful owners, a balancing of accounts." Christs work is to restore the world, God's estate, to its rightful owner, and to place man and the world back under God's law, and, in the process, to balance all accounts, restoring to God what belongs to God, and casting out of God's presence and realm the reprobate into Gehanna or dung heap of creation.
Acts:3:22-26: 22beginning from the baptism of John to that day when He was taken up from us, one of these must become a witness with us of His resurrection.” 23And they proposed two: Joseph called Barsabas, who was surnamed Justus, and Matthias. 24And they prayed and said, “You, O Lord, who know the hearts of all, show which of these two You have chosen 25to take part in this ministry and apostleship from which Judas by transgression fell, that he might go to his own place.” 26And they cast their lots, and the lot fell on Matthias. And he was numbered with the eleven apostles.
Here is a powerful address to warn the Jews of the dreadful consequences of their unbelief, in the very words of Moses, their favourite prophet, out of pretended zeal for whom they were ready to reject Christianity, and to try to destroy it. Christ came into the world to bring a blessing with him. And he sent his Spirit to be the great blessing. Christ came to bless us, by turning us from our iniquities, and saving us from our sins. We, by nature cleave to sin; the design of Divine grace is to turn us from it, that we may not only forsake, but hate it. Let none think that they can be happy by continuing in sin, when God declares that the blessing is in being turned from all iniquity. Let none think that they understand or believe the gospel, who only seek deliverance from the punishment of sin, but do not expect happiness in being delivered from sin itself. And let none expect to be turned from their sin, except by believing in, and receiving Christ the Son of God, as their wisdom, righteousness, sanctification, and redemption. -Matthew Henrys