The jumping in ETG is all about setting the horse up for success. It's ultimately the horses decision whether it will jump or not, and its behaviour on the jumping course is affected both by personality and stats.
The jumps can only be jumped when in a canter.
It's best to approach a jump straight on, and not from an angle. Jumping from an angle makes it harder for the horse to clear the jump without knockdowns. But horses are also less willing to jump from an angle. The higher the jump, the more difficult will it be for the horse to jump at an angle. The higher Mentality stat, the more likely is the horse to be willing to jump from a bigger angle, while low Mentality horses will be very particular about jumping straight-on.
When approaching the jump, the rider needs to give rein and jump together with the horse. A rein release bar will appear, with an marker that indicates the optimal timing for the rider to initiate the jump and give rein. Depending on the player's timing, they can either get a perfect (green), good (yellow) or bad (red) score on the rein timing. The higher the jump, the harder it gets to get the timing perfectly. Horses with good Mentality will not care as much if they only get a good, or even a bad rein timing -- if their Mentality is high enough, they will attempt the jump anyways. However, if the Mentality is low, the horse is likely to demand a perfect timing, or else it will refuse.
The higher the jumps, the higher the demand on Mentality for both rein release and jump angle.
The canter speed affects how the jump will turn out. The faster the canter, the longer (but lower) the horse will jump. This is good for wide jumps such as triple bars and liverpools. On the contrary, when in a slow canter, the horse will jump shorter, but more upwards to get maximum jump height. This is good for high jumps such as verticals. Once a jump has been completed, an icon of the next jump will appear on the screen. The player can use this information to adjust their stride in preparation of the next jump.
It's important to remember that all horses are different. A calm horse will strive to slow down, so it can easily break down into trot when the rider was aiming for a slow canter, while a high spirited horse might run away in an extended canter. The more the player gets to know their horses, the easier will it be to foresee and counter these behaviours (or turn them into an advantage!)
Finally, all horses have a limit to their potential and not every horse is built to become a top tier show jumper. Through training and skillful riding, all horses can compete and reach some success on the jumping course, but once the potential is reached, that horse cannot progress into higher tiers.