In the same way that this commission is to every believer, not just to ministers, the command to baptize is also for every believer. It is proper for any believer to administer the ordinance of water baptism.
Jesus commands us to baptize in the name of the Father, the Son, and the Holy Ghost, yet there is no recorded instance where the believers did so. Instead, the instances of baptism that are recorded in the book of Acts, show people being baptized in the name of Jesus only.
Because of this noticeable difference, doctrines have arisen that teach there is no Trinity, and that unless water baptism is administered in the name of Jesus (only), with faith in the water baptism, a person cannot be saved. This is a false teaching that has led many people astray.
Water baptism is an outward witness of the inner change that has already taken place. In Acts 10:44-48, Cornelius and his friends were filled with the Holy Ghost and spoke in tongues, proving that they were already born again before they were baptized in water.
The harmony between the commission of Jesus and the practice of the early church is simple. When we baptize people in the name of the Father, Son, and Holy Ghost, we are baptizing them in the name of Jesus because ‘in him (Jesus) dwelleth all the fullness of the Godhead bodily’ (Col. 2:9).
Therefore, baptizing in the name of either, the Father, Son, and Holy Ghost, or the name of Jesus, is correct as long as Colossians 2:9 is understood.