There is a two-fold purpose in regard to the baptism of repentance for the remission of sins. The one who was baptized with water played a role much greater that just submitting to that rite. Let us look at the following passage and then we will determine the meaning of the word "repent":
"In those days came John the Baptist, preaching in the wilderness of Judaea, And saying, Repent ye: for the kingdom of heaven is at hand. For this is he that was spoken of by the prophet Esaias, saying, The voice of one crying in the wilderness, Prepare ye the way of the Lord, make his paths straight" (Mt.3:1-3).
In this instance the Greek word translated "repent" means "to change one's mind for better, heartily to amend with abhorrence of one's past sins": Mt. iii.. 2 ; iv. 17 ; Mk. i. 14" (Thayer's Greek English Lexicon).
In Barnes' Notes on the Bible we read that "There are two words in the New Testament translated 'repentance,' one of which denotes a change of mind, or a reformation of life; and the other, sorrow or regret that sin has been committed. The word used here is the former, calling the Jews to a change of life, or a reformation of conduct" (Notes on the Bible, Albert Barnes, Commentary at Matthew 3:2).
In Vincent's word study we read "Repentance, then, has been rightly defined as 'Such a virtuous alteration of the mind and purpose as begets a like virtuous change in the life and practice'" (Vincent's Word Studies, by Marvin R. Vincent).
At Matthew 3:3 we read of John the Baptist "preparing the way of the Lord" in regard to the kingdom which was at hand. In what way was he preparing the way?
It was necessary for the people to have a change of mind in regard to their sinful life style in order "make ready a people prepared for the Lord " so that they "might serve Him...in holiness and righteousness" (Lk.1:17,74-75).
This does not imply in any way "faith" in the gospel but instead things in the "moral" sphere and not the "spirtual" sphere. Let us look what Peter said to the nation of Israel on the day of Pentecost:
"Repent therefore, and turn back, that your sins may be blotted out" (Acts 3:19; ESV).
Here, the Greek word translated "turn back" means "to turn back morally, to reform...Acts iii. 19" (Thayer's Greek English Lexicon).
So the word "repent" as it is used in regard to the "baptism of repentance for the remission of sins" had nothing to do with anyone believing the gospel.
"And now why tarriest thou? arise, and be baptized, and wash away thy sins, calling on the name of the Lord" (Acts 22:16).
In his commentary on this verse Sir Robert Anderson writes, "The Apostle records the words which Ananias addressed to him (Paul) at his conversion: 'Arise and be baptized, and wash away thy sins, calling on the name of the Lord'...His meaning is clear: 'Arise and be baptized, and turn away from your evil courses, calling on His name'" [emphasis mine] (Anderson, The Bible or the Church? [London: Pickering & Inglis, Second Edition], 230-231).
That is the meaning of the words "ye washed yourselves" and "cleanse ourselves" in the following verses:
"And such were some of you: but ye washed yourselves, but ye were sanctified, but ye were justified in the name of the Lord Jesus, and by the Spirit of our God " (1 Cor.6:11; RV, Marginal Note).
"Having therefore these promises, dearly beloved, let us cleanse ourselves from all filthiness of the flesh and spirit, perfecting holiness in the fear of God " (2 Cor.7:1).
When anyone submitted to the rite of water baptism he was pledging to change his life for the better. That is why John the Baptist said the following to the Pharisees and Sadducees:
"O generation of vipers, who hath warned you to flee from the wrath to come? Bring forth therefore fruits befitting repentance " (Mt.3:7-8).
John the Baptist was telling them that since they were pledging to change their way of living then they should exhibit a real change in their lives. Here the Greek word translated "repentance" means "the change of mind of those who have begun to abhor their errors and misdeeds, and have determined to enter upon a better course of life, so that it embraces both a recognition of sin and sorrow for it and hearty amendment, the tokens and effects which are good deeds...Mt. iii. 8" (Thayer's Greek English Lexicon).
Therefore we can understand that those who were baptized with water were to have a change of mind in regard to their sinful lifestyle and were pledging to change their life for the better and in return the Lord would cleanse them from all their sins.
Next, we will look at Acts 2:38 and see the role which the Lord played in regard to the remission of sins.