The use of land and/or buildings for the keeping of horses for commercial or recreational purposes will usually require planning permission. The erection of buildings to shelter or exercise horses or to store their food will also usually require permission, along with operations such as installing a surface for an exercise area, lighting or the siting of a caravan in connection with the use.
There are limited circumstances where permission may not be required. These include where the main purpose of the land is for the genuine grazing of horses (i.e. they are being fed off the land rather than by imported food and the land is not being used as a recreational/exercise area) as this could be considered to be an agricultural use. However, buildings and other structures will still need permission.
A stable within the curtilage of a house might not need permission, subject to the size of the stable, its position on the land, and the horses being kept as pets by the occupants of the house.
You will often find that your local authority has specific policies in their Local Plan regarding new stables/livery/riding school development. It is often carefully controlled due to impacts on the local highway network, residential amenity including noise and the visual amenity of the countryside. It is usually advisable to approach your local planning authority with a pre-application enquiry to get their formal advice on relevant issues and the likelihood of permission being granted.