Go and Sin No More

The villain has either been defeated or acknowledges the error of his ways. He surrenders to the hero, expecting to be killed or punished in some other way. Instead, the hero lets him go, telling him to repent for his crimes by doing good deeds. This usually turns the villain into The Atoner and send him on a Redemption Quest.

Trope Namer is The Bible, in which Jesus releases an adulteress rather than stone her to death, as was common at the time.

Examples of Go and Sin No More include:

Anime and Manga

Done several times in Rurouni Kenshin.
Goku of Dragonball Z tries this on Friesa twice.
This happens in the Nuts Fujimori manga adaptation of Fire Emblem: Seisen no Keifu. In this adaptation, one of the early bad guys, Kinbois of Verdane, is portayed as deep down not being evil, just someone influencable who was following the wrong path. Sigurd notices this, and lets him live on the promise he will atone for his sins. Kinbois is genuinely moved by this mercy and promises. This turns out to be crucial, as Kinbois' brother Jamuka later fights Sigurd because he believes, in a misunderstanding, that Sigurd has killed his beloved brother in battle: after learning the truth and seeing Sigurd's generosity, Jamuka joins his party.

Fan Works

Invoked by Doug Sangnoir in the final chapter of Drunkard's Walk II, after infecting four (temporarily deactivated) boomers with a nanobot/organism that will free them from control by GENOM, who quotes the Trope Namer at them.

Film

In Gandhi a Hindu comes to Gandhi as he's partaking in a fast and says he (the Hindu) is going to Hell, because he killed a Muslim child in revenge for Muslims killing his son. Gandhi tells the man to to repent by finding a Muslim boy whose parents have been killed, and raise him. But here's the kicker: he's to raise the boy as a Muslim.

Literature

Done by R. Daneel to the corrupt police chief in Caves of Steel by Isaac Asimov. Instead of arresting the chief for the murder of a pro-robot, pro-space travel scientist, Elijah and Daneel tell him to rally support for robots and space travel among the Luddite "Medievalists" he worked with.
Subverted in Wizard's First Rule. Richard forgives his brother Michael for everything he did to him. However, Richard still executes him as he does not forgive him for his sins against others.

Live-Action TV

In Doctor Who, the Doctor makes a Sontaran work as a nurse.
In The Sarah Jane Adventures, an alien who had been kidnapping children to use as child soldiers realizes that the war his people were fighting had ended without his knowledge. He allows Sarah Jane Smith to take his life, but she refuses. The alien then promises to return to his homeworld and bring back the humans he had abducted.
Power Rangers:
- In general, whenever Tommy has a say in how to handle a formerly-evil Sixth Ranger Traitor or Token Evil Teammate (both Kat and Trent come to mind) he will be the one arguing strongest for clemency. After all, he was the original Evil Sixth Ranger and was given a second chance by the original team.
- From Power Rangers in Space, when Astronema figures out that she's Andros' sister, she turns herself in to the Rangers, and they - minus Andros - have to discuss how to handle this in private. She wonders if they're planning to dump her into a black hole, and Andros doesn't help matters by saying there's one nearby. To her surprise, they pardon her rather quickly.

Oral Tradition, Folklore, Myths and Legends

The Trope Namer is a story in The Bible.
If you identify "the hero" in the page description with God, you'd have a fairly accurate representation of the Christian doctrines of repentance and salvation.

Video Games

In several Mass Effect games, you can either kill an enemy as a Renegade decision or let him go for Paragon points. Most of the time it will usually work out in Shepard's favor to spare the character.
In the The Legend of Zelda CDI Games, after Duke Onkled betrayed the King of Hyrule by siding with Ganon, he gets off by being told to "Scrub all the floors in Hyrule."