Maura is right, there are a few different styles of correct seat. Personally, I find the most secure seat for all occasions is the eventing type seat, but I disagree that there is any amount of gripping. Gripping and tension are what get you off of a horse in a big spook. Instead, it is a method of weight distribution that puts an equal amount of weight lightly against the saddle and the horse from two inches below your crotch to the bottom of your foot.
Your leg is lightly on your horse (except for where it naturally curves away between the bottom of your calf and your foot) but not gripping or cueing. You are not standing heavily in the stirrups, but there is enough weight in there that if your horse were to suddenly lurch to one side, your stirrups would not come off your feet. To a casual observer, it looks like you are sitting on the horse, but your weight is only lightly in your seat and not pressing down like a sack of potatoes.
From this position it is easy to either stand up off the horse's back if you run into some rough ground or use your seat more strongly in the saddle for forward encouragement. I call it the "ski position" because you are bending your joints to keep your legs balanced under you whether you are standing straighter above the saddle or sitting down on top of it. It is good for riding strong horses on open country.
However, if you are showing in any discipline you need to learn what seat is required, whether the chair seat of saddleseat, the draping seat of dressage or the full butt seat of Western.