Repentance has several different meanings depending on the context.
- Repentance is a feeling of regret or sorrow; when you wish you had never done something.
Example: Genesis 6:5-6 "And it repented the LORD that he had made man on the earth, and it grieved him at his heart." God is sorry that he created humans, they have become so wicked.
- Repentance is a turning away, to cease. Example: Revelation 2:16 "Repent; or else I will come unto thee quickly, and will fight against them with the sword of my mouth." The Holy Spirit is warning the church of Pergamos to stop their fornication.
There is command to believe: "Repent ye, and believe the gospel" (Mark 1:15). How can anyone believe in Christ if they don't change their thinking? No one naturally sees themselves as a sinner; when you repent, you know and realize you're a sinner. You change from thinking "I'm a basically good person." to "I'm a sinner; I need Jesus." That is the third definition of repentance.
In Luke, the people saw the Galilaeans killed by Pilate, and asked Jesus if this happened because the Galilaeans were terrible, wicked people. "And Jesus answering said unto them, Suppose ye that these Galilaeans were sinners above all the Galilaeans, because they suffered such things? I tell you, Nay: but, except ye repent, ye shall all likewise perish." The people needed to stop thinking of others as bad and themselves as good.
Yes, they felt remorse for their sin, (1st definition), but this is not a requirement of believing in Jesus. They couldn't have stopped sinning, (2nd definition), this only comes after you are saved. They needed to repent (3rd definition) to believe and be saved.