There are two categories in which you can show a jumping horse,
Hunters and Jumpers

HUNTER
The United States Equestrian Federation, or USEF, states that hunters are judged on
performance and soundness, an even hunting pace, manners , jumping style
together with faults and the way of moving over the course. Horses must be shown
over a minimum of eight obstacles at the required height.

Faults are scored according to the judge's opinion, and depending on severity, may
be considered minor or major faults.
Showing an obstacle to horse
Missing a lead change
Kicking out
Spooking
Jumping out of form
Jumping into corners of obstacles

Major Faults
Knock down of any part of an obstacle
Refusals
Trotting while on course
Bucking
Stopping for loss of shoe or broken equipment
Circling while on course
Dangerous jumping

Elimination
Three refusals
Off course
Jumping a fence before it is reset
Bolting from the ring
Fall of horse or rider

The following may or may not be considered as faults, depending on their severity  
and frequency
Light rubs
Swapping leads in a line
Late lead changes
Excessive show of animation
Adding or eliminating a stride in a line

Hunter obstacles must simulate those found in the hunting field such as natural post
and rail, brush, white board fence or gate, hedge, oxer, etc. A ground line is
recommended for all obstacles.

JUMPERS

Jumpers are scored mathematically, based on penalty faults incurred between the
starting line and finish line. Penalty faults include disobediences, falls, knockdowns,
touches and time penalties.

Jumper Course There must be at least one change of direction in every course
including jump-off

Courses
At least once combination must be included in every course
Every course must contain a minimum of eight obstacles
At least three obstacles out of the first eight must be spread obstacles

Elimination
Three refusals
Off course
Fall of horse or rider
Jumping an obstacle before it is reset, or without waiting for signal  to proceed
Starting before judge's signal to proceed
Failure to enter ring within one minute of being called
Failure to cross the starting line within 60 seconds after signal to proceed
Exceeding the time limit
Actions against a horse deemed excessive

Riding a hunter or a jumper is very technical and requires that a person be able to
do many different things all at once and all at the right time. Balance, timing, trust in
your horse, and leg strength and sight all play key rolls when jumping a horse. It
takes years of practice/training to make it all come together. Once must be able to
fall and get back on. Jumping is not for the timid.
Hidden Treasure Equestrian Farm