The hard part of a saddle is called a saddle tree.
The saddle and bridle of a horse is called its tack. The tack includes the bridle, saddle, saddle pad, gierth, and sometimes a martingale.
It is called the Cantle.
Individually they are called a saddle and bridle, together they are called tack.
A saddled horse
When you are riding a horse without a saddle it is called bare backHope this was helpful!:)