I am a parent judge and have no prior personal experience in debate. I have judged at a number of tournaments over the past year and have judged each of the debate events.
The strength and persuasiveness of your arguments are the most important, but specifically I watch for three things: organization, performance and clash.
Organization: I want to be able to track all of your arguments. Make it obvious when you are beginning a new argument or rebutting one of your opponents. Signposting is helpful.
Performance: Act confident, speak clearly, stand up straight and look up from your computer. Do not speak too fast. Use enunciation and inflection in your voice. If you're speaking so fast that I can't follow what you're saying your argument is wasted.
Clash: This is a debate event and I want to see debate! I watch for rebuttals of specific arguments, I also watch how well you handle yourself during crossfire. Do you ask intelligent questions, when answering questions are you able to respond knowledgeably and confidently?