As an Instructor that primarily teaches groups of children there are really only two things that get on my nerves.
1. Blasting down a beginner trail. If I have my train of new guys coming down, DO NOT GO FLYING RIGHT NEXT TO THEM AT FULL SPEED. You're a great skiier and you know you're not going to hit them, hell you actually passed them with 5 feet of space in between you, thats fine and dandy but when my 5 year old starts crying because he thinks he almost just got hit, or worse panics and falls/hits someone else its just unacceptable. If you want to go that fast please stick to more advanced trails, I promise that we keep the new guys on the beginner ones.
2. Being a bad role model. No, they're not your kids or your responsibility, but don't be screaming obscenities at your friends across the trail when you're in earshot of the little guys. Also please don't be going off-piste / making your own jumps etc in front of them. I once had a guy come down through the woods (not allowed at my mountain), hit a little kicker and come flying up and go OVER my student. If the kid had been half a foot taller, or the skier hadn't been going fast enough my kid would have gotten a ski in the head/neck. The skier then had the gall to try to chew me out for 'not keeping the landing clear'. I calmly asked my co-instructor to take the class down the hill and wait at a landmark for me, and then preceded to rip this particular skier apart. That was the only time I've felt the need to report a particular skier/boarder to a patroller.