Riding a horse English style refers to the type of saddle, bridle, and tack the horse is ridden in, as well as how the rider rides and sits on the horse.
English tack is different from Western tack in that it is much simpler, and usually lighter weight. Riding a horse English style includes a rather flat saddle with no saddle horn or large swells, and a plain bridle. Unlike most Western bridles, English bridles have a "cavesson". This is a strap that goes over the horses nose.
How you ride a horse English style is relatively similar to how you ride a horse Western style. However English riders ride with their lower leg more underneath their body, and english riders ride with two hands on the reins instead of one. Contact through the reins is taken on the horses bit, and there is rarely slack in English riders' reins. This is not more harsh on the horses mouth, because English riders use a different type of bit that is usually much less harsh than the Western style curb bit is.
You may ask "how do I ride a horse English style?" and the best way to learn how to ride a horse English style is by taking English riding lessons from a local riding instructor or riding stable. This will allow you the opportunity to learn how to ride English style safely and become knowledgeable about English style riding.
This is one type of English saddle:
This is a photo of a horse being ridden English style:
