Over fences the judge is looking for the most smooth and effective course first, and body position second. (I actually thought it was the other way around until I attended a judge's clinic a while ago). That means you need keep your horse balanced and supple, keep your horse on a solid rhythm (unless told to do otherwise), nail every single distance, nail every lead change, tight balanced turns, straight lines, etc.... in as subtle a ride, in as correct a position as possible.
At the shows around here the lower level eq courses really aren't too different then a hunter course with a roll back and/or a bending line, pretty much every set on the same length of stride. When you get higher up they will start shortening or lengthening the lines so you need to make a decision of how to best ride the line and adding some trickier turns. You want to get as tight of a turn as you can, taking the inside turns, but ONLY if you can do it flawlessly. It's better to take the outside turn and doing it well rather then to do the tricker inside turn and mess it up. however, if both turns are executed properly the judge will give more points to the inside turn. In a medal ride off they'll give you a test that could involve things like a counter canter, a certain number of changes of lead, halt, etc etc.
if you want to see some really nice eq rounds look up Maclay Finals on youtube. A few of the past recent winners were Brianne Goutal, Zazou Hoffman, and Jessica Springsteen. Can't for the life of me remember who won it this past year, unfortunately I wasn't able to go!