Acts 2:38 records the Apostle Peter’s words on the day of Pentecost, “Peter replied, ‘Repent and be baptized, every one of you, in the name of Jesus Christ for the forgiveness of your sins. And you will receive the gift of the Holy Spirit.’” This was a strong affirmation by Peter that “there is no other name under heaven given among men by which we must be saved” (Acts 4:12). Being baptized in the name of Jesus indicates an understanding by the person being baptized that Christ is the Savior.
Christian baptism is also in the name of the Father, Son, and Holy Spirit (Matthew 28:19). Being baptized in this manner simply means we are identifying ourselves with the Trinity. We belong to the Father, are saved by the Son, and indwelt by the Spirit. This is similar to how we pray in Jesus’ name (John 14:13). If we pray in the name of Jesus, we are praying with His authority and asking God the Father to act upon our prayers because we come in the name of His Son, Jesus. Being baptized in the name of the Father, Son, and Holy Spirit is being baptized in identification with them and their power over and in our lives. Jesus Himself specifically tells us to baptize “in the name of the Father and of the Son and of the Holy Spirit” (Matthew 28:19).
In the Book of Acts, new believers were baptized in the name of Jesus (Acts 2:38; 8:12; 8:16; 10:48; 19:5). It is, however, essentially the same thing—Jesus, the Father and the Holy Spirit are one (John 10:30; Acts 16:7). Per Jesus’ own instructions, believers should be baptized in the name of the Father, Son, and Holy Spirit, but as the book of Acts proves, baptizing in the name of Jesus is also done. The bottom line is that the name/names in which we are baptized is not as important as the recognition that baptism identifies us with the death, burial and resurrection of Christ, our Savior. We are buried with Him and risen to walk with Him in newness of life.