Hopefully, you are always going to choose when you get off of your horse. But, 'unscheduled dismounts' happen, and it is best if you know how to handle the situation, whether it's you or someone else who falls off.
If you are going to spend a lot of time around horses, it is a good idea to take a first-aid course. This will prepare you for basic emergencies and give you the confidence and skill to handle things like lacerations, strains/sprains, and bruises.
Trail Riding in a Group
Ideally, when riding in a group while trail riding, try to be aware of the skill and confidence level of the other riders and the temperament and training of the horses. You then need to plan your route and set your pace with the least capable horse and rider pair in mind. Overwhelming a horse or rider by going too fast, traveling over rough terrain, or going too far and fatiguing are invitations to trouble. Many falls and other precarious situations can be avoided by applying some common sense in planning your ride.
If, however, one rider and their horse do 'part company' you will have to take a quick assessment of the situation. Sometimes riders are lucky enough hold onto the reins, dust themselves off and remount. If they're a bit shaky after the fall, it's a good idea to dismount your own horse and steady theirs as they mount up. However, in the case of a really bad fall, you need to look after the rider first. Even if the horse leaves the scene, you will have to wait until you check to see if the rider is okay before retrieving the horse. If the rider needs first-aid or medical attention, see to this first. Carrying a cell phone while riding can be a huge help, as can letting someone know what your planned route is. This way help and emergency crews can find you quickly.
If the Horse Leaves
Only after the rider is looked after is it time to look to the horse. If the horse has left the scene, it may simply find its way home. If not, the loose horse may seek companionship and safety near other horses or even other livestock. If the horse isn't found quickly, notify authorities, providing clear photos and a description. Begin your search at the place the horse was last seen. Notify anyone living in the vicinity and leave your contact information with them. Searchers should have halters and leads as the loose horse’s bridle may have come off.
Riding in an Arena
When you ride in a ring or arena, you need to be as aware of green horses and/or novice rider combinations as you would be when riding on the trail. Arena etiquette is also important when there are several people riding at the same time. Stables should have emergency numbers such as fire and ambulance, and have the physical address of the barn posted. If someone does fall off, all other riders should halt their horses and dismount if needed.
Falls in an arena can spawn 'chain reactions' among the horses. If a loose horse runs around excitedly, the other horses may become excited too, causing more problems. The instructor or someone knowledgeable in first-aid should attend first to the rider, and someone will need to catch the loose horse. Only after the rider is looked after and the horse is under control should the others mount up and continue their ride.
Getting Back On
If the rider and horse are not injured, the rider should be encouraged to re-mount the horse and continue the ride. It's understandable to have your confidence shaken, sometimes badly, after a fall, but re-mounting quickly after the fall will help the rider mentally push these emotions aside. Talking about what happens and why and how (if possible) the fall could have been avoided can help everyone learn from the situation, turning a negative into a positive.