An example: My uncle, who I suspect was a psychopath, spent much of his life counting on my father, among others, to bail him out of trouble. See, there was a family rule in my father's family that everyone was supposed to take care of this person. My father followed this rule, doing things like lending my uncle money that he knew he'd never see again, paying my uncle's bail so he wouldn't have to spend a few nights in jail, forgiving my uncle for various wrongs he did to my father, and so forth.
Then, my father had his niece and nephew over for a visit, and his niece revealed that her father, my uncle, was sexually abusing her. My father's behavior to his brother changed immediately - he turned him in to the police, did his best to ensure no contact between my uncle and his children, and for the rest of my uncle's life he had as little contact as he could with my uncle. (Which, for the period of time I can remember, consisted of seeing him once at a funeral, where my father did not acknowledge his brother and turned away when his brother tried to talk to him.)
My uncle never seemed to get that it was over. He kept trying to mend the relationship, to convince my father to forgive him. At court at one of the custody hearings, he apparently ordered his lawyer to say that he just wanted everyone to get along. Even 15+ years later at a relative's funeral he tried to talk to my father, hoping to patch things up. After all, my father had always forgiven him before.
You see this clearly in a lot of fiction. There are many heroes who engage in transgressive behavior, like beating people up or in some cases killing them. But if they cross a certain line, they will cease to be considered a hero. (Admittedly, rules in real life are usually stricter than in fiction, because people don't have to face the real-life consequences of the actions such as making a mistake, setting a precedent, etc.)
Dexter, for example, despite being a serial killer, manages to be a hero, because he has certain lines he never crosses. The big one is that his violence to others is proportionate to the violence they have engaged in. He only kills people who have killed multiple people themselves. The times that he faces foes who have never killed, such as his girlfriend's ex-husband (a physically and emotionally abusive drug addict), he deals with them through methods other than murder. Even Doyle, who found him out and was planning to expose him, he was not able to bring himself to kill.
Another line he never crosses is that he never kills children. Once, he contemplated killing a budding serial killer teenager, but the plot conspired to ensure that he changed his mind. This is important because children, even when they have done wrong, are not considered to be as responsible for their actions as adults are. And because children are designed to elicit maximal levels of empathy - all immature mammals have certain physical and behavioral traits that mammals are instinctively predisposed to consider 'cute', in order to elicit nurturant behavior from them. Dexter's caring behavior towards children, therefore, serves as a signal to viewers that despite being a serial killer he is still a hero.
There are other uncrossable lines for many people, such as betrayal or sexual violence, but those are the main ones. Furthermore, each empath is somewhat unique in this regard - while the big things tend to be universal, every person draws the line slightly differently. For example, people who hold strong views about tolerating diversity may lose all sympathy for someone who expresses a racist, sexist or homophobic view, while other people think it's wrong but not unforgiveable. Some people see infidelity as unforgiveable, others do not. My father did not consider reporting his brother to the police to be crossing the line (provided he'd done something bad enough, which he had), but his family did. Often if you ask a person what they consider unforgiveable, they can name specific things, though sometimes people don't know that they find something unforgiveable because it hasn't occurred to them that anyone would do that.
And an important point: this line tends to be mostly the same regardless of whether the wrong was done to them, to someone they care about, or even to someone they don't know or don't like. The one exception is that many people tend to make excuses for the wrongs done by people they care about and therefore may require a higher standard of 'deserving victim' if they liked the victim. But for the most part, it doesn't make much of a difference. My father reacted similarly to my uncle abusing his own children as he'd have reacted to my uncle abusing me (which he did not have the opportunity to do, fortunately). He'd have been more emotionally distraught about me being abused, but he'd have taken exactly the same measures. Furthermore, if he had not known the children at all before discovering they were being abused, he'd likely have reacted similarly.
* Non-psychopath - includes NTs plus people with other brain differences that are unrelated to psychopathy.