quantam savvy does a good job describing it simply. Their word for parellis "friendly game" is "no yield contact". That's where you do something with the horse like swinging a rope around your horse and expect the horse to standstill & relax, and not yield from (or push into) that pressure. Next game being the "contact yield' ("purcupine game") that's where you want the horse to yield to pressure it can feel. Like pressure on the reins/seat/legs and on the ground picking up their feet and anything that involves steady pressure on the horse. Then there's "no contact yield" ("driving game") this is where you want the horse to yield from more rythmic pressure that doesn't necessarily have to physically touch the horse, any kind of "suggestive" pressure. An example of this is a lunge whip.
Those are the basic parelli games, the rest are just combining some of those basics to achieve purposes like what most would call lunging ("circle game", excersize and learning to maintain gait & direction), pushing the horse sideways ("sideways game", good for flexibility) "squeezing" the horse between 2 scary objects, usually the human and another big object, while expecting him/her not to get claustrophobic and rush ("squeeze game", good for self confidence in the horse)
The other game is the one in the middle called the yo-yo game, one "yo" being forward movement and the other "yo" being backward movement. Priorities of the yo-yo game are getting equal sensitivity to forward and backward aids, and depending on which "yo" you consistently finish on, it builds more go than whoa, or more whoa than go in the horse. Great for horses that are "hard to rein in" or horses that need constant nagging to go forward. Next priority being straightness, the point of that is that when our horses are going forward or backwards we want exactly that, minimum lateral movements, good for horses that constantly bend to one direction and not the other.
Despite what the parelli's say, horses don't play these 7 games with eachother. I have yet to see horses play the circle game or the sideways game in a herd situation. Horses only play the first 3 games with eachother (friendly/no yield, porcupine/contact yield, driving/no contact yield)