Answer:
In the US, a Livery is the term for a place where you can rent a horse or horse and cart. A Boarding Stable is the term used for a place to keep your horse.
If you have nowhere to keep a pony/horse, or can't look after it yourself, you can pay a stable yard. This is called keeping your pony at livery. Full livery is when everything is done for you, all you do it turn up and ride. Part livery is when you some of the time looking after your horse/pony. DIY livery is when the yard provides a field and stable, and you do the rest yourself..