Hunter- usually in an arena with 8 or so jumps. You have a set course and it's all about the look of the horse and rider, how they jump, the striding, and poles down. Basically- look good doing it :)
Here is a person doing hunters: http://images.onlinecommercegroup.co...mper-Horse.jpg
Jumpers- it's all about time and rails down. If you get the fastest time and don't have rails down, then you win.
Here is a person doing jumpers: http://www.hunterjumpernews.com/wp-c...ne%20Photo.jpg
Cross Country- you jump immobile objects such as logs, ditches, banks, corners, water, etc. on a set course. There is an optimum time and the person with the lowest score after the last phase of eventing (stadium/showjumping) wins. If there is a tie, the person closest to the optimum time wins.
Here is a picture of someone doing cross country: http://regardinghorses.com/wp-conten...ss-country.jpg