The short answer is that baptism is an outward expression of a person surrendering his will to the Lord. Spiritual baptism refers to the inward decision to surrender that will, and thus become a part of the body of Christ, receiving the Holy Spirit. Although the phrase "spiritual baptism" does not occur in the Bible, we will assume you are referencing a "baptism with the Holy Spirit," specifically separate from the act of literally immersing in water. According to Strong's Concordance, the word baptism means to immerse or submerge or to overwhelm. Today, the term is frequently associated with the ritual of water baptism, so it can be difficult to distinguish between the spiritual and water baptisms. However, spiritual baptism is not a literal immersing in water, but rather a submerging of your will into God’s will.
This spiritual baptism is further explained as a baptism into Jesus’ life of self-sacrifice culminating in death. Romans 6:3 reads, "Do you not know that all of us who were baptized were baptized into his
This spiritual baptism also means that one receives the Holy Spirit (“baptize you with the Holy Spirit,” Matthew 3:11). This power of the Holy Spirit transforms into Christ-likeness. (Rom. 12:1, 2), and unites us into one spiritual body. 1 Cor 12:13,27 "For in one Spirit we were all baptized into one body…now you are the body of Christ and individually members of it." The passage in chapter 12 goes into more detail about how individual members work together to create the body of Christ, but the idea is that even though individual members are each unique and fulfill a different role, they are all part of a greater, unified purpose. Ephesians 4:4-5 illustrates this unity well, "There is one body and one Spirit…one Lord, one faith, one baptism," indicating that upon baptism, we are dedicating our lives to serve the Lord and have that as our only true purpose and focus.
It is our understanding that this spiritual baptism occurs first inwardly, and can then be represented outwardly with a water baptism. In other words, water baptism itself does not change a person, but rather the inward decision and begetting of the Holy Spirit which is represented by a water baptism changes them. In1 Peter 3:20-21, Paul explains this point, "baptism…now saves you, not as a removal of dirt from the body but as an appeal to God for a good conscience, through the resurrection of Jesus Christ," indicating that the literal removal of dirt from the water baptism only represents a removal of sin through the Spirit. Acts 8:35-38 also teaches that water baptism is a representation of the inward, heart commitment. Here, Philip preaches to a Eunuch who then wishes to be baptized in water. Verse 37 reads, "And Philip said, If thou believest with all thine heart, thou mayest." This indicates that the heart condition must exist (internal decision) before the outward water baptism can take place.