Typically, two sets of judges visit your pit at the regionals. They come by usually before lunch to interview your team. They like to talk to the students about the team. This where Al mentioned the mentors should stand back and let the students present the information about their team. Some judges may also return to your pit for additional information after lunch. Try to always have some students in the pits who are comfortable talking with judges.
One set of judges will be asking question about your robot, things like did you make use of innovative controls or design techniques, how you built the robot, or how you programmed it. They will also be looking at the overall quality of your robot, pit, and cart to see if it is clean, organized, and expresses a complete package.
The other set of judges will be asking questions about your team. They will be looking for things like a complete and expressive image for your team, your team spirit, business plans, how you organize your team, etc.
The judges then return to the judging room to discuss the teams and identify which teams will be getting the various awards. The judged awards that are debated during this process includes the Engineering Inspiration award.
The Regional Chairman's Award is a bit different. First, it is only available for teams in their second year and beyond, by definition of the award, rookies are not eligible, although they do have Rookie Inspiration and Rookie All Star which parallel the RCA and EI awards. The RCA award is based on a submitted essay, a short video on DVD, and the RCA interview done at one regional with a small part of the team (up to 3 team members IIRC).
So, if you want to see all the awards that are available at a regional, check out this link:
http://www.usfirst.org/sites/default...0%20Awards.pdf
For some great tips in regards to judging, check out the Awards Resources on the NEMO site:
http://www.firstnemo.org/resources.htm