We are up to Mishnah 7 in chapter 8 of Tractate Sanhedrin, the Mishnah that concludes the chapter. This Mishnah deals with cases in which it is permitted to kill a person even without bringing him to Beis Din, in order to prevent him from committing a severe transgression. This is the classic and well-known case of the 'Rodef' (pursuer).
The text of the Mishnah: "V'eilu shematzilin osan b'nafshan" - and these are the ones whom we save with their lives. The scenarios listed here are situations in which it is permitted to take the life of the potential transgressor, in order to save him from committing the transgression.
There are two foundational principles for this permission:
Saving the pursuer from himself - he is about to commit a terrible transgression that will destroy his soul, and some of these transgressions are among those whose punishment is sekilah (stoning).
Saving the victim - the pursued is about to suffer a horrific injury, and therefore it is permitted to save him by stopping the pursuer.
The practical order of action: First, one must instruct the pursuer to stop his action. If he does not listen - it is an obligation to stop him, and ideally without killing him, if possible. If there is no other way, it is even permitted to kill him and take lethal action against him. Unlike the previous Mishnah, this is not merely a permission but an obligation and a mitzvah incumbent upon any bystander to stop the pursuer from carrying out his plot.
What are the transgressions in which we save them with their lives?
"Harodef achar chaveiro l'horgo" - one who pursues his friend to kill him. This refers to Reuven who is pursuing Shimon to kill him. One must instruct Reuven to stop, and if he does not listen - one must do everything necessary to prevent him from committing the murder, even at the cost of killing Reuven himself.
"Achar hazachur v'achar na'arah hame'orasah" - after a male and after a betrothed maiden. This is one who pursues a man or a woman in order to rape them. We are always talking about a pursuer who is a man, and the pursued is forbidden to him. Saving the pursued from the rape is a mitzvah in itself, and therefore a bystander who sees this must stop the pursuer, even if it entails killing him.
And what are the transgressions in which we do not save them with their lives?
"Abal harodef achar behemah" - but one who pursues an animal. This is one who pursues an animal to commit bestiality with it. Even though there is a potential liability of sekilah here, we do not kill him for it, but rather bring him to Beis Din after the fact.
"V'hamechalel es HaShabbos" - and one who desecrates the Shabbos. This refers to one who is about to desecrate Shabbos by performing forbidden labors and the like.
"V'ha'oved avodah zarah" - and one who worships idols. This refers to one who is about to engage in idol worship.
In all these cases, the Mishnah states: "Ein matzilin osan b'nafshan" - we do not save them with their lives. One must try to dissuade the person from committing the transgression, but if it is impossible to stop him - it is forbidden to kill him. Rather, we bring him to Beis Din, and Beis Din does what is required of it.
And if Beis Din is unable to execute him for whatever reason, he is still liable to death at the hands of Heaven. Regardless, we do not kill a person outside the framework of the law, unless we are saving not only the person from himself, but also a potential human victim from the terrible and horrific crime that is about to be committed against them.
In summary: In this Mishnah we learned the law of the 'Rodef' - an obligation incumbent upon every person to stop the transgressor who is about to murder or rape, even at the cost of taking his life. We noted the two foundational principles of this permission: saving the pursuer from the destruction of his soul, and saving the pursued from harm. In contrast, regarding one who pursues an animal, desecrates Shabbos, or worships idols, where there is no human victim, we do not save them with their lives, but rather we bring the transgressor to Beis Din.