What Happens if I Fall Off?

The answer to this question really depends on the rules of your national federation. If you check any rule book, it should probably come under "Falls". Some federations will not allow you to remount in the show ring due to safety considerations while others will allow you to remount and ride out of the ring.

If you fall off during a class, Equine Canada rules state that you are eliminated and can not get back on the horse in the show ring:

ARTICLE G103 FALLS
After being eliminated for a fall, the rider may not remount in the ring. Any rider who does remount in the ring after a fall will be eliminated from all classes in that ring for the remainder of the day. A rider who remounts after a fall and takes another obstacle will be disqualified from the remainder of the competition.

If you fall off and you are able to keep a hold of your horse, lead your horse out of the ring at the walk. If you fall off and your horse runs away, either stay where you are awaiting medical assistance or help to catch your horse. Once he has been caught, you or someone else can lead him out of the ring. Once you have left the show ring, you can remount in the warm-up area and continue to ride.

Keep in mind that the rule refers to falls in which the horse is eliminated. If the fall occurs before the start timers in a jumper class, the horse is not eliminated and the rider can remount (Disclaimer: I have not seen this happen in several years so the general interpretation of the rule might have now changed):

FEI ARTICLE 203 BELL

1.2. [...] Disobediences, falls etc. occurring between the signal to start and the moment the competitor crosses the starting line in the correct direction, are not penalized.

You may be given a leg up by a third party (i.e. a member of the jump crew) provided you have not yet crossed the start line:

FEI ARTICLE 225 UNAUTHORIZED ASSISTANCE
1. Any physical intervention by a third party between the crossing of the starting line in the correct direction and the crossing of the finishing line after jumping the last obstacle, whether solicited or not, with the object of helping the competitor or his horse is considered to be unauthorized assistance.

For hunters, here is the specific EC rule:

ARTICLE G407 THE SCORING OF HUNTERS

5. Horse and/or rider falling while in competition incurs elimination, and a rider may not remount in the ring. A competitor is considered to have fallen when, either voluntarily or involuntarily, he/she is separated from his/her horse, which has not fallen, in such a way that he/she touches the ground or finds it necessary, in order to get back into the saddle, to use some form of support or outside assistance. A horse is considered to have fallen when at the same time both its shoulder and quarters have touched either the ground or the obstacle and the ground.

For clarity, here is the definition of a fall:

FEI ARTICLE 224 FALLS
1. A competitor is considered to have fallen when, either voluntarily or involuntarily, he is separated from his horse, which has not fallen, in such a way that he touches the ground or finds it necessary, in order to get back into the saddle, to use some form of support or outside assistance. If it is not clear that the competitor has used some form of support or outside assistance to prevent his fall, the benefit of doubt must be given to the competitor.
2. A horse is considered to have fallen when the shoulder and quarters have touched the ground or the obstacle and the ground.