Been tied up, unable to comment here. Threads gone in several directions... but...
First... Forms, and their purposes. Let's start by noting that the question was asked in the General Martial Arts forum, and seemed broadly aimed at forms. I say that because there are different sorts of forms in different styles. There are forms that are meant to teach particular principles or concepts. There are forms that encapsulate a system's techniques and principles. There are forms done purely for esthetics; they're beautiful or athletic, and that's what they're supposed to be. There are forms that are meant as moving meditations, or spiritual aids. There are forms that are meant to build strength and physical ability (yeah, a yoga sequence is a sort of form...). There are partnered forms that record exactly how a technique is supposed to be done and work against someone -- and solo forms done without anyone else. So... if you don't like competitive forms, or your system doesn't include them -- cool. Don't bash people who do or systems that do. Train your way, and let others train theirs. I'm not a fan of the XMA stuff -- but I can't knock the athleticism and physicality of the competitors!
Which leads to judging... There are two broad types of tournaments: closed/style specific, and open/non-restricted. If I'm judging a closed tournament in my system, I know what those forms are supposed to look like. I know what the stances are supposed to be, where you can add theatricality or showmanship and where it's hiding poor technique. And I can tell whether a pause is dramatic or confusion... So they get graded one way. I look for proper execution of our principles, proper stances, the proper form, etc. I know that even though one form may be shorter than another, it's also more intricate or complex while another form is incredibly physically demanding. But in an open tournament, I don't know the forms. Across the Japanese Karate (and many of the Okinawan Karate, too), there are a number of fairly common kata, even if they aren't exactly the same from style to style. But in the various Chinese arts, there can be different forms in the same named art... And then you get us oddball styles like Bando... So how do I judge fairly in open tournaments? I look for balanced footwork, solid stances, clean techniques that show focused targets, and I look for whether or not the performer seems to know the form. It's usually pretty obvious if they get lost; pauses in weird places or for too long, techniques that just don't connect, etc. Length and apparent complexity figure in, too -- longer or harder forms are likely to score better in one division than shorter/simpler ones. Where it gets really hard is when you're trying to score forms that use very different principles against each other -- especially if you don't understand the principles. Watch a hard style judge try to assess a soft style form -- or vice versa!! -- sometime... They sometimes just don't connect it...
So... advice to a new judge, in an open tournament? Do your best to be fair and to be consistent in how you judge. If you're way off base from the other judges, speak to them about what they're seeing that you're not, good or bad. Personally, especially for new judges, I like to have 5 judges, and discard high & low scores. But at least if you're internally consistent with yourself -- your scores will be fair for all the competitors.
Advice for a competitor? Know your form. Know where you can add dramatic touches, and use those moments to help distinguish yourself. Understand the rules for a competition; are the judges told to look for strict adherence to the exact form, or is their room for variants? Have a second go-to form in your back pocket, in case something happens that you get told to do a different one (like a run off between two competitors). Never let the judges know you made a mistake; they'll either already know, or if they don't -- you can fool them. (I know someone who won a grand championship despite inventing about 1/2 the form because he got lost... but the judges didn't know!) Afterwards, if you get a chance, thank the judges and ask them politely what you can do to compete better next time.
