Repentance, here, comes from the Greek μετανοέω (metanoeō), which means to have a change of mind. It is not a conscience decision to think differently, but it is what the Holy Spirit does to make a sinner realize his condition (Jn. 16:7-11). It is what it takes to be saved: trusting in Jesus Christ (Acts 16:31). Repentance is similar to the Gestalt Switch which takes place when one looks at the drawing by Toulouse-Lautrec, of the ugly old woman who suddenly "changes" into a beautiful young girl. When one hears the Gospel message, and believes that his sin was the reason God sent His only begotten Son to die on the cross to pay for his sin, repentance occurs. He doesn't "turn over a new leaf." He doesn't determine to live the rest of his life according to a religious teaching so that he can earn salvation through good works. He turns from being the one making life-choices, to wanting to please God out of pure gratitude for what His Son did for him!
In Luke 13, Jesus was speaking to Jews. He was telling them that the tragedies did not signify that those who perished were especially evil, but that they too were sinners and subject to death at any time. Notice He said all of them needed to repent, and that without repentance, they would all perish. The Apostle Paul wrote that all men sin, and the Apostle Peter wrote that God wants "all to come to repentance" (Rom. 3:23; 5:12; 2 Pet. 3:9). As I understand the passage, Jesus was saying that the men who had died, had not repented of their sins, and therefore, they died in their them. John 8:24 says, "I said therefore unto you, that ye shall die in your sins: for if ye believe not that I am He, ye shall die in your sins."
Repentance is what happens to you when you come to faith in Jesus Christ! You have a change of mind. Have you turned from controlling your life according to what pleases you, to allowing the will of God to guide your decisions?
Don't allow "a Pilate" or "a tower" to end your life while you are living in sin!
"Behold, now is the accepted time; behold, now is the day of salvation" (2 Cor. 6:2)!