Many at Pentecost had been convicted of their sins. In order to bring spiritual comfort into their experience Peter gave three promises to them.
First, he assured them that if they repent of their sins and confess Jesus by faith, they will receive from God the forgiveness of all their sins. They will receive an immediate free and full pardon. In a moment, they will turn from those under the bondage of condemnation to those enjoying the liberty of the forgiven. What a great blessing pardon from God is! Delivered from condemnation, and delivered from the fear of condemnation.
Second, Peter says that they will receive the gift of the Holy Spirit. Jesus had promised his disciples that when the Spirit came after the Ascension, he would be in them as well as being with them. Of course, the indwelling of the Holy Spirit is a massive subject, and no one can fully describe what it involves. In saying that the converts would receive the Spirit, Peter was promising them many blessings: there is the blessing of purity as they are sanctified by the Spirit; there is the blessing of power as they are enabled to serve Jesus by the Spirit; there is the blessing of the fruit of the Spirit as they receive these gifts from Jesus through the channel of the Spirit. To have the Spirit is, as we know, to have the presence of God.
Third, Peter reminded his hearers that they would come into a covenant relationship with God if they believed in Jesus. This is what Peter means when he says that ‘the promise is for you and for your children and for all who are far off, everyone whom the Lord our God calls to himself.’ In the Old Testament period, the children of parents in Israel received privileges from God because he had made commitments to their forebears. This does not mean that the children of converts, whether from Jewish families in Israel or Jewish families in the Dispersion, would automatically become Christians. But it is a reminder that members of Christian families are very near to gospel blessings, which means that Christian parents can plead this relationship when they pray to God.