"...I kept back nothing that was helpful, but proclaimed it to you, and taught you publicly and from house to house, testifying to Jews, and also to Greeks, repentance toward God and faith toward our Lord Jesus Christ." - Paul (Acts 20:20-21)
There are many Christians who do accept repentance as a part of salvation (as well they should), but they view repentance as only "a change of mind" (from the Greek word metanoia). Repentance is not only a change of mind, I think, but it's also a change of direction that results in a change of action -- not merely an acknowledgement of sin, but also a turning from sin (and self) and to God.
Repentance is a complete surrender of the heart, mind, and will (a supernatural work of God). It's a proper response to the conviction of sin. That said, repentance is NOT "cleaning up yourself first" before you can come to God for saving. Not only is that impossible, but that is works and works cannot save anyone! (See: Are good works enough? 8 Reasons Why We Can't Work Our Way To Heaven) I like what one preacher said: "The call to Repentance in salvation is not 'clean up your act first for Jesus to accept you and save you.' You don't get good to get saved; you get saved to get good. Repentance is a deep conviction that you can no longer continue on the road that you are on, a road of destruction...a road of self rule or self righteousness, which leads to eternal ruin. And with that conviction, you turn to Christ in faith believing that He is the only Savior able to set you free from the penalty and power of sin through His substitutionary death and resurrection from the grave."
In short, God brings the sinner to Himself through repentance and faith in Jesus Christ in an initial act known as justification (the sinner is made right with God). Subsequent to justification is sanctification (the process of becoming more like Christ); this is a continual process throughout the earthly life of the believer. All of this is by God's grace. It is God's grace that saves and it is God's grace that sanctifies!
[There is no repentance without faith and there is no faith without repentance. Faith without repentance (easy-believism) is a false gospel and only produces false converts (See: False Converts Need to Be Warned and How Does the Bible Define a False Convert?).]
Genuine saving faith is always accompanied by repentance. A true saving faith is penitent faith (repentant faith): Godly sorrow. A false non-saving "demon" faith is impenitent faith (unrepentant faith): worldly sorrow. Godly sorrow (contrition) over sin produces a broken and contrite heart which compels the sinner to apologize to God and plead for mercy. In contrast, worldly sorrow (attrition) over sin is really a "non-apology" to God and more of a "sorry I got caught" mentality and maybe a hatred for the consequences of sin, but not necessarily the sin itself. Worldly sorrow over sin leads to death. Only Godly sorrow over sin produces repentance that leads to salvation: Life. (See: Godly Sorrow, Saving Faith vs Demon Faith, and also Mike Gendron's presentation below).
"What is Biblical Repentance and Is It Required For Salvation?" - Proclaiming the Gospel TV with Mike Gendron (WVW TV)
Description: 2 Corinthians 7:9-11 describes Godly sorrow produces repentance unto salvation. Repentance is taught in the Bible over and over. Jesus commanded His disciples to go forth and preach repentance. What exactly is Biblical repentance? Why is repentance so seldom preached, explained, or discussed in modern churches? Why must we discuss and explain repentance when sharing the gospel? What is the connection between the Holy Spirit and repentance? Could that lack of including repentance in the gospel message explain why there are so many false converts? Watch here.
(PDF) Repentance and the Gospel by Mike Gendron
Repenting Unrepentant Repentance - Mike Abendroth - NoCo Radio Part 1
Repenting Unrepentant Repentance - Mike Abendroth - NoCo Radio Part 2