The 5 main types of competitions for English riding are dressage, hunters, jumpers, cross country, and halter showing. Each type of competition is very difficult, and it takes a very seasoned and experienced rider to successfully compete or show.
There are also many other types of riding, however, they are not classified under the English style of riding. Some of these include carting, roping, poles, and barrel racing.
Videos from you-tube: http://www.youtube.com/watch?v=sRT79xoLrGo , www.youtube.com/watch?v=SnOYxVYU26s , www.youtube.com/watch?v=o8UpLM57ulQ , https://www.youtube.com/watch?v=o8UpLM57ulQ
Images from: http://en.wikipedia.org/wiki/Lusitano , http://www.flyingwfarms.com/Awards/2006_Year_End_Award_Winners%20.htm , http://www.horseshowsbythebay.com/horse-shows-101 , http://en.wikipedia.org/wiki/Cross-country_equestrianism , http://www.worldhorsewelfare.org/Event/Brechin-World-Horse-Welfare-Horse-and-Pony-Show
Images from: http://en.wikipedia.org/wiki/Lusitano , http://www.flyingwfarms.com/Awards/2006_Year_End_Award_Winners%20.htm , http://www.horseshowsbythebay.com/horse-shows-101 , http://en.wikipedia.org/wiki/Cross-country_equestrianism , http://www.worldhorsewelfare.org/Event/Brechin-World-Horse-Welfare-Horse-and-Pony-Show