A long time ago, in a galaxy far, far away, I was a moderator on a pretty active political forum. It was a bit of a chore, but I felt it was worth my time to facilitate such discussions. I also learned a lot about how both moderators and users behave, about what works and what doesn't. In more recent days, I've been involved in more online discussions than usual, on forums with a variety of approaches to moderation. What has become clear to me is that many people who have only now started having conversations this way have no idea how to do so, and that many moderators were unprepared for current levels of intensity. Here are some suggestions.

My first (meta-)suggestion is: don't try to play moderator if you aren't one. Some sites (e.g. Hacker News) even have rules forbidding this, and for good reason. It never works. In 35+ years of online discussion I've come to realize that everyone thinks they are the "voice of reason" entitled to interpret the purpose of the forum and the rules of engagement for others. Everyone is wrong. When you attempt to play moderator for two people who never asked for it, you're placing yourself above both. That is a form of escalation (look it up), not moderation. The results are predictable. Usually, the disputants will just expand the conflict to include you. If your phrasing seems judgmental or dictatorial enough, they might even forget their original differences and ally against you. Others who feel equally entitled (I'm using that word deliberately) but interpret the situation differently are also likely to jump in. Before long the meta-discussion will be more of a mess than the original. Requests for moderation are fine, but as soon as you start "laying down the law" for others you're only likely to make things worse.

Here are some other (shorter) tips for actual moderators.

• Do realize that, as a moderator, you are over-exposed to people's worst comments. Selection bias is an occupational hazard. Unless you actively guard against it, you won't be able to distinguish prolific net-positive contributors who have a few bad moments from full-time trolls. Don't let yourself be fooled.
• Do realize that people will use the moderation system to further personal or political goals. People rarely report their friends for anything short of a major transgression, or even at all. Many will routinely report their enemies for very minor transgressions, or even for nothing at all. Moderate according to need, not demand. Don't let yourself be used.
• Don't rely on history. Don't consider a person's history as a "good" or "bad" person to decide those borderline cases, especially since your perception of that is likely to be skewed (see two previous bullets). Doing that and then throwing it in their face where everyone can see is far worse. (Disclosure: part of the reason I'm writing this is because of a recent incident on HN where a moderator did exactly that.) It's a form of insult, which is never good for the conversation even from a regular user and is made even worse by your official role. It comes across as - and effectively is - harassment.
• Don't insult users. Don't say their comments were "worthless" or "nasty" etc. Say those comments were against the rules, but keep your personal opinion of their quality to yourself.
• Do make sure the rules are clear. For a current example, if you're going to say "no politics" then you'd better define what's considered political. Some people think the mere mention of COVID-19 statistics or scientific findings is political, and don't even get me started on masks. If the rule you're supposed to enforce isn't clear, then you'll just have more damaging meta-discussions.
• Don't invite or indulge in discussion of moderator actions. That just exacerbates every other problem mentioned here. Keep it short and to the point. Cite the rules if you feel you must, then move on.
• Do act on patterns. Wait, what? Didn't I just say not to consider people's histories? That was for purposes of deciding how (or whether) to react to a single comment. Evaluating the totality of a user's contributions is a different matter. If the sum of a person's contributions is sufficiently net-negative over sufficient time and warnings, get rid of them. You will very likely get flak for that, but you'll be doing the most important part of your job.
• Don't get in arguments with other moderators, and if you do FFS don't let the users see it. Publicly disputing or reversing another moderator's decision is among the worst things you can do in this role.

Being a moderator is hard. It's hard work, and it's emotionally draining. Doing it wrong only makes it harder, which can lead into a real death spiral as moderators get increasingly frustrated and do even worse while users lose faith in the entire system and leave. But if you do it right, civil discussions leave everyone happier and better informed. It's worth the effort to learn how.