How to be a good adjudicator in debate?

Are you talking about a formal debate, or something more relaxed?

In formal debates, you look at a variety of things. Here is the list from the American Parliamentary Debate Association. It suggests that the judge become a blank slate, and act as if they know nothing other than what the debaters present. Any feelings you have about the debaters should be put aside. Then they judge based on the facts provided by debaters, and the persuasiveness of arguments and rebuttals made. It has nothing to do with any knowledge you had previously, and only to do with how well the debaters do.

In informal debates, of course, you can take into account things you know. You can decide whether debaters are lying or misrepresenting the facts based on your knowledge base. You can include things such as whether you like the debaters or not.

In formal debates, the judge does not get to raise questions, nor do they give the debaters any advice on how to debate. If you are confused, that counts against the debater. It is not your job to ask them for clarification.

In informal debates, there are no rules, so you can engage the debaters however you want. If you want to ask questions, you can. If you don’t understand something, you can ask for clarification.

Political debates are a different animal, altogether. They are somewhere inbetween formal and informal debates. But they tend to have a moderator who is different from a judge. The moderator asks questions and gets things going and asks for clarifications if they want to. Sometimes they allow the debaters to ask each other questions. In political debates, the public is the judge, and they get to judge however they want to.

I feel that in any circumstance a good adjuicator should be able to identify and discredit fallacious arguments….. Outside of that. See above.

The article pointed to by @wundayatta mentions the importance of taking notes. Debates have a pro and a con side. Make an outline of the arguments provided by the pro side, along with your evaluations. Then make a list of the rebuttals to those arguments given by the con side, in addition to any new arguments that they make. Good debaters will help you with this by outlining their arguments before giving specifics and by summarizing their arguments at the end.