What does it mean to repent? Is repentance necessary for salvation today?

The Greek word translated repentance means "to change one's mind." It is a changing of mind about God, about sin, and about one's self. It goes much farther than being sorry for our sin. The "goodness of God" leads us to repentance (Romans 2:4), and it is produced in us by the Holy Spirit through the Word of God. We are made to feel how terrible our sin is before a holy God and this in turn leads us to turn away from our sin.

Repentance is a vital part of the gospel message. Our Lord Jesus said "Except ye repent, ye shall all likewise perish" (Luke 13:3). The apostle Paul preached "repentance toward God, and faith toward our Lord Jesus Christ" (Acts 20:21). Real "believing" on the Lord Jesus Christ then includes both repentance and faith. See also Luke 5:32, Acts 17:30,31 and 2 Peter 3:9.