Steeplechasing was invented in Ireland and is so called because riders would race across country, as straight as possible, from one church steeple to another.
The term has now been formalized to refer to fairly short distance races that are run over obstacles. In Europe, all steeplechases are run over brush fences on an enclosed track, although ditches and water are common. In America, steeplechases are run on open courses over solid timber fences.
Point to pointing is used in Britain and Ireland to refer to amateur steeplechases in which the horses are required to be actively used in foxhunting (these days, no foxes are involved).
Horses jump a water obstacle in an English steeplechase. Source: Paul Holloway via Wikimedia Commons.
The most famous steeplechase races include the English Grand National, the Maryland Hunt Cup and the Velka Pardubicka (considered the world's most dangerous horse race) in the Czech Republic.