Having good arena etiquette is not just polite, it is also mainly for safety. With a ring packed full of sometimes 20 horses or more, it is important that everybody is on the same page and obeys traffic laws. Not following the rules of the ring should get you pulled out of the schooling ring until you re-think your actions, after-all, nobody wants to get hurt.
At most horseshows and riding stables, it is common for a slower horse to work the rail and the faster horse to pass on the inside. Only once have I encountered a woman who tried to pass on the inside and it did not work well. Typically the arena operates like the highway, drive in the right lane or outer lane and pass on the inside. This is very common sense and is easy to remember, not to mention it is habit for many folks and horses.
Whether you are schooling at a show or working in an arena at a stable, you need to be considerate about which direction you're working. Typically you work the same direction as everybody else in the ring. This prevents collisions. Many times the ring director will call "reverse" and all in the ring will change direction. If the arena is not too crowded, you may circle at half the arena or stay to one end if you feel you must work your horse the opposite direction for a bit. Otherwise you can ask the group, "Mind if we reverse?".
Etiquette becomes even more important if you are working over fences such as in the hunter/jumper world. In a schooling ring, plowing headlong down a line of fences without giving anyone notice can cause a bad collision. Commonly when it is clear and your turn, the rider calls which line of jumps they want to ride. For instance, "Heads up, vertical, or "Heads up oxer." This lets people know the line is occupied and people around know to stay out of the way.
Also, if your horse is not jumping well and is giving you trouble, it is rude to monopolize a jump. Don't park your horse in front of the jump and yell at it or swat it repeatedly. After 2 or 3 attempts, move on and let others school the fence. Problems are to be worked on at home, not at a show. Sometimes there is even a small practice jump out side the arena in a warm-up area, you may try that.
There will be times at any horse show unfortunately when someone will fall off. Many times when a rider is down, it is polite to slow or stop your horse along the rail. This will prevent the rider if injured from being trampled. Also since there will likely be a loose horse in the ring, if the other horses stand still, it will not excite the other horse as much.
Having good ring etiquette will make other riders appreciate you and will keep you and your horse safe. Be courteous, be polite, be watchful and most important, be safe.