"If a burglar is caught in the act of breaking in, and is struck and killed, it is not considered an act of murder."
The idea is, if the burglar is breaking in to a house at night, when there is a reasonable expectation that the resident will be at home (as opposed to during the day) it's a logical conclusion that the burglar intends harm to the residents, and that killing him would be an act of self-defense. This verse is the basis of the Talmudic concept that a rodef - one who is pursuing another with the objective intent to kill, must be killed to save the life of the innocent victim. And in some circles of thought, this principle is applied to the issue of abortion, where the unborn can be seen as a rodef, threatening the life of the mother.
Another important principle established from millenia of Jewish scholarship is "when in doubt, refrain."
As for forgiveness, this is the subject of the Rosh HaShannah ("Head of the Year) and Yom Kippur (Day of Atonement) holidays that mark the beginning of the Jewish year.
The Yom Kippur service pertains to asking forgiveness from G-d for our failures to live up to G-d's expectations, our sins of omission, sins of commission, known or unknown in the past year. The slate is figuratively wiped, so to speak, and a Jew can start off again with a new resolve to live as G-d commands. But an important thing here is that sins committed against another person must be resolved with the other person, before forgiveness can be asked of G-d.
