I believe that today there is only one literal baptism usually referred to as Christian baptism or "baptism into Christ."
There is only one.
1. Ephesians 4: 4-6 - "There is one body and one Spirit, just as you were called in one hope of your calling; one Lord, one faith, one baptism; one God and Father of all, who is above all, and through all, and in you all."
The question is asked: "If there is only one baptism, why does the writer of Hebrews refer to "baptisms"? Hebrews 6:1, 2 - "Therefore, leaving the discussion of the elementary principles of Christ, let us go on to perfection, not laying again the foundation of repentance from dead works and of faith toward God, of the doctrine of baptisms, of laying on of hands, of resurrection of the dead, and of eternal judgment." (NKJV)
There are several baptisms mentioned in the Bible, but discounting those that were only symbolic and those that were temporary, there remains only one baptism practiced in the church today. It is immersion in water in the name of Jesus Christ for the remission of the sins of those who come to Jesus, repentant of their sins and trusting in His death on the cross to take away their sins.
Various Baptisms in the Bible
Baptism of John
Mark 1:4 - "John the Baptist appeared in the wilderness preaching a baptism of repentance for the forgiveness of sins."
Acts 18:25 - "This man had been instructed in the way of the Lord; and being fervent in spirit, he was speaking and teaching accurately the things concerning Jesus, being acquainted only with the baptism of John."
Acts 19:4 - "And Paul said, 'John baptized with the baptism of repentance, telling the people to believe in Him who was coming after him, that is, in Jesus.'"
The baptism of John was temporary and no longer to be practiced after Christ gave his life on the cross.
Baptism in Fire
Matthew 3:7-12 - "But when he saw many of the Pharisees and Sadducees coming for baptism, he said to them, 'You brood of vipers, who warned you to flee from the wrath to come? Therefore bring forth fruit in keeping with repentance; and do not suppose that you can say to yourselves, 'We have Abraham for our father'; for I say to you that God is able from these stones to raise up children to Abraham. And the axe is already laid at the root of the trees; every tree therefore that does not bear good fruit is cut down and thrown into the fire. As for me, I baptize you with water for repentance, but He who is coming after me is mightier than I, and I am not fit to remove His sandals; He will baptize you with the Holy Spirit and fire. And His winnowing fork is in His hand, and He will thoroughly clear His threshing floor; and He will gather His wheat into the barn, but He will burn up the chaff with unquenchable fire."
The baptism in fire is symbolic of God's judgment on the unrepentant.
Baptism into Moses
1 Corinthians 10:1-2 - "For I do not want you to be unaware, brethren, that our fathers were all under the cloud, and all passed through the sea; and all were baptized into Moses in the cloud and in the sea."
When Israel left Egypt, crossing the Red Sea, they were encompassed by water - the cloud over them and the Red Sea around them. This is a symbolic use of the word "baptism" in several details, not only them being surrounded by water (though passing on dry land). It is indeed a prophetic type of our experience. As they were freed from their slavery and came into the relationship with Moses as their leader, so we, in baptism, are freed from the bondage of sin and come into a relationship with Jesus as our Lord.
Baptism of Suffering
Matthew 20:20-23 - "Then the mother of the sons of Zebedee came to Him with her sons, bowing down, and making a request of Him. And he said to her, 'What do you wish?' She said to him 'Command that in Your kingdom these two sons of mine may sit, one on Your right and one on Your left.' But Jesus answered and said, 'You do not know what you are asking for. Are you able to drink the cup that I am about to drink?' They said to Him, 'We are able.' He said to them, 'My cup you shall drink; but to sit on My right and on My left, this is not Mine to give, but it is for those for whom it has been prepared by My Father.'"
In Matt. 26:39 - He prays to the Father to "let this cup pass from me; yet not as I will, but as thou wilt." In John 18:11: "the cup which the Father has given me, shall I not drink it?" The "cup" and "baptism" Jesus spoke of here was a symbolic way of talking about the terrible things he would soon suffer as he offered himself as our sin sacrifice. It is symbolic language.
In Acts 1:5, Jesus said to them "For John baptized with water; but you shall be baptized with the Holy Spirit not many days from now."
The baptism in the Holy Spirit was what Jesus did with the Spirit on the day of Pentecost.
John 15:26 - "But when the Helper comes, whom I will send to you from the Father, that is the Spirit of truth, who proceeds from the Father, He will bear witness of Me."
On the day of Pentecost, Jesus poured the Spirit out on "all flesh" (see Acts 2:17). The baptism in the Holy Spirit was a once for all time historical event. The effects remain but the Spirit has already been poured out on all humanity.
Baptism into Christ
Mark 16:16 - "He who has believed and has been baptized shall be saved; but he who has disbelieved shall be condemned."
Gal. 3:27 - "For all of you who were baptized into Christ have clothed yourselves with Christ."
Romans 6: 3 - "Or do you not know that all of us who have been baptized into Christ Jesus have been baptized into His death?"
There is now one baptism. All the others are either symbolic uses of the word meaning "immersion" or historical events that do not need to be repeated.