The Promise 2:38-39
“38) Peter said to them, “Repent, and each of you be baptized in the name of Jesus Christ for the forgiveness of your sins; and you will receive the gift of the Holy Spirit. 39) For the promise is for you and your children and for all who are far off, as many as the Lord our God will call to Himself.””
‭‭Acts‬ ‭2‬:‭38‬-‭39‬ ‭
The word “repent” means regret accompanied by a true change of heart toward God.  It is a change of mind and heart.  It includes a change of morals and direction in life. It is an aligning with Gods morals, character and holiness from our own sinful ways.
The importance of baptism is that it is an outward expression of something inward.  Peter said this baptism in the name of Jesus Christ was for “the forgiveness of your sins.” By contrast, those baptized by John the Baptist were most likely baptized in the Name of the Father strictly as an expression of repentance.  Something very beautiful is starting to take shape. 
When Jesus gave The Great Commission in Matthew 28:19, He talked about baptizing in the name of the Father, the Son and the Holy Spirit.  Here we see the Trinity. This threefold baptism started in earnest after the Day of Pentecost. I believe it involves a unique relationship with God through each member of the God-head. 

Repentance is best pictured by the story of the prodigal son in Luke 15:11-32. He had a genuine change of heart and mind.  He humbled himself and went back to his father with no demands or conditions.     True repentance was the basis of everything else that happened in the story.  As a result, he received the following three things: 1) Acceptance illustrated by the embrace from the Father, 2) Forgiveness illustrated by new clothes, and 3) Power illustrated by the family ring.

This is more than coincidental. These lines up with baptism in the name of the Father, the Son and the Holy Spirit.  They represent three very important aspects of salvation that are an immediate reality at the moment of salvation, but need to be worked out in depth over time in the life of the believer. 

Let me say it another way.  Your growing relationship with the Father will enable you to embrace total acceptance and family identity.  Your growing relationship with Jesus will enable you to enter into the full realization of forgiveness and freedom.  Your growing relationship with the Holy Spirit will enable you to experience power and anointing. Relationship with each member of the God-head is important.  

I see all three of these promises being offered in Peter’s message.  Receiving the gospel results in a relationship with God unknown in Old Testament times.  Only through the gospel can people can be fully reconciled to God. 

This Promise is for everyone who receives the gospel. Let me emphasize; “16) For God so loved the world, that He gave His only begotten Son, that whoever believes in Him shall not perish, but have eternal life. 17) For God did not send the Son into the world to judge the world, but that the world might be saved through Him.” (John‬ ‭3‬:‭16‬-‭17‬) 

Daily Bible Commentary 
By Terry Baxter: Cofounder of GoServ Global