What is the common name for the part of the saddle that the rider sits on?

The part of the saddle that the rider sits on is commonly referred to as the seat. The seat is specifically designed for comfort and support during riding, allowing the rider to maintain balance and stability. It is typically cushioned and shaped to fit the contours of the rider’s body, enhancing the overall riding experience.

While the tree is the framework that shapes the saddle and gives it structure, and the cantle is the raised back part of the saddle, which provides additional support for the rider’s back, these components do not serve as the surface where the rider sits. Stirrup is another important part, providing a place for the rider’s feet, but again, it is not where the rider sits. Therefore, "seat" is the most accurate and common term to describe the part of the saddle designated for the rider to sit on.