Hunt seat is your basic english hunt riding.
If you've been riding hunter jumper- you've been riding hunt seat. That's the style of riding you do in an English (hunter) saddle. Like Western riding- you have reiners and Western Pleasure and gaming but it's all still western (stock seat) riding.
It's basic equitation on an english horse, shown at the posting and sitting trot, and with extensions of the trot and canter. There are different types and headsets. There is the QH/Paint/Stock type:
And then Morgans and Arabs and other hot breeds:
(This is what I personally do)
Saddleseat is a whole different ball game!
Ridden in a cutback saddle on (usually) American Saddlebreds (also Morgans, Arabs, Tennessee walkers and other gaited breeds) it's the horses with weighted shoes and huge flashy movement. The rider (ideally) sits upright, straight backed in the saddle with a 'chair seat', or with the legs more forward. They are placed over the center of the back, off the shoulder, to maximize front end movement. There are classes for gaited and non-gaited horses. They are ridden in double bridles.
Tennessee Walker (Gaited Saddleseat)
Hunt saddle: (obviously)
Saddle seat (cutback) saddle: