What it entails depends on what you mean by competitive trail ride (CTR). ACTHA hosts rides called trail challenges. These are eight mile courses that take around two and a half hours to complete. You are judge on your horse's performance across natural(ish) obstacles along the trail. It's like a trail class on an actual outdoor trail.
These types of rides are good for pleasure type distance riders who just want to have some fun once in a while. Since the courses are so short, it doesn't require much conditioning.
On the more endurancey end of things are NATRC rides. I do these, so I can tell you more about them than ACTHA. On a NATRC ride, you are given a map of about 20 miles of trail, a pace (3 or 4 mph), and a time (e.g. 5 to 5 and a half hours). Along the trail are points (A, B, etc.) to guide you along and to use as milestones to make sure you're on time. The aim is to ride 20 miles of trail in the allotted amount of time -- neither too fast nor too slow.
As you're fighting with time, judges are hiding along the trail, watching and waiting. You have two cards: horse and horsemanship. Each is judged separately. The vet judge has power over your horse card. This judge is looking for a well conditioned (checks out sound and has good P and Rs)and well mannered trail mount. The horsemanship judge has power over your horsemanship card. This judge will look for a safe campsite, good equitation, and your trail manners, along other things.
The basic aim is to survive two 20 mile days and check out at the end with a horse who could turn around and do 20 more.