I'll concur with the others, but add "even-handedness" as a key trait and the lack thereof as a potentially fatal flaw. For example, if:
Someone that everyone likes and respects is being a jerk towards someone else who totally deserves it
. . . you have to see it as:
Someone that everyone likes and respects is being a jerk towards someone else who totally deserves it
....
The mods in this forum have for the most part been fair, which sadly meant a couple of valued popular forum members have been permanently banned. OTOH, there is at least one member who posts almost nothing but insults and ridicule and I take it many people think it's funny and harmless. That's unfortunate since this particular member's constant ridicule is not funny or harmless if you are the butt of the joke, especially since the people who are the butt of jokes can't respond without breaking rule whatever number it is.
So I suggest that being able to recognize when a joke is not harmless and everyone isn't laughing would be a good quality for a mod. And being able to recognize when ganging up on someone is going on is important.
I think a good mod would be able to recognize that the discussions where tempers flare are qualitatively different from the posters who ridicule and purposefully try to hurt people's feelings and belittle them. There is always going to be accidental belittling that sometimes happens when people forget themselves and blurt something out maybe they regret later, like when you say something someone said is stupid. Or a person's posts can tend to sound rude but there isn't a pattern of malice there, just an argumentative debater or someone who gets angry but ridicule and insults are not the goal, rather they are reactive. These flare ups need to be stopped when they get out of hand, but they differ from the posts where the intent is to belittle someone and an active argument is not behind it.
It can be hard sometimes to see the difference between an argument where tempers flare and the more bully-like behavior of constantly making fun of someone, which often includes recruiting cheerleaders to make fun of the person the bully is making fun of. I've seen many members here that have been ridiculed on a regular basis. And there are several members who do a lot more ridiculing than simply getting angry in a debate.
Being purposefully nasty and hurtful should not be tolerated. Losing it in an argument in a forum where people are naturally disagreeing with each other OTOH, just needs to be defused. The problem is, if you don't stop the ridicule, it grows and appears acceptable. If you let arguments get out of hand they appear acceptable. I think the mods have been good about defusing arguments. I wouldn't want that job for sure. But the mods here have IMO, been less quick to intervene when ridicule occurs. It makes the forum unpleasant.
Putting people on ignore works for many situations. And there are some posters who seem immune to constant ridicule. The politics forum, for example, has a number of people who continually ridicule each other and for the most part the ridiculees don't appear to be bothered by it. I'm not sure what a mod should do about this. On the one hand, no one may be bothered. On the other hand, allowing the behavior to go on sets a precedent that can be a problem later. So this is a difficult problem for a mod.
And balance is just as difficult. The BAUT forum (Dr Plait's forum) is waay over moderated IMO. It evolved that way when Phil, who was good at managing the board, had to delegate to mods. Or maybe it was because the forum grew too big. Anyway, now so many discussions just don't get interesting because the mods seem to stop all the passion behind a topic.