Can you define this Great Game and what each kingdom want to achieved?

Well, the Great Game is the term I use to describe “that long epoch between the assimilation of the Andals and the coming of the dragons” when the “Kings of the Reach
warred constantly with their neighbors in a perpetual struggle for land, power,
and glory. The Kings of the Rock, the Storm Kings, the many quarrelsome kings
of Dorne, and the Kings of the Rivers and Hills could all be counted amongst
their foes (and ofttimes amongst their allies as well.)”

In terms of what the countries wanted, the prize of the Great Game was the conquest of all of southern Westeros, or at least as much of southern Westeros as possible in the hopes of becoming a continental hegemon that could overawe those parts of Westeros it couldn’t conquer outright.

No one quite succeeded in that ambition, although many tried, in no small part because one of the rules of the Great Game is that the moment anyone looked to be winning, everyone else would gang up against them. Hence when the Reach had conquered all of the Stormlands save for Storm’s End itself, “the King of the Rock swept down upon the Reach in his absence, forcing him to lift his siege and hurry home to deal with the westermen. The broader war that followed involved three Dornish kings and two from the riverlands.” Likewise, when Arlan III conquered the Riverlands, “the Dornish came swarming over the Boneway to press them in the south, and the Kings of the Reach sent their knights forth from Highgarden to reclaim all that had been lost in the west,” and eventually the Ironborn took it off them. And just before Aegon landed, it looked like Harren the Black might be the next up for the dogpile. 

The somewhat annoying thing about the Great Game, as I was just discussing with @goodqueenaly, is that we don’t have enough information from the sources to chart the whole 5,000 year period: we have a good bit of info about the early Great Game (from the Andal Conquest to the time of Lancel IV, Gyles III, and Torrence Teague), and we have a good bit of info about the late Great Game (from the time of Arlan III to Aegon’s Conquest), but the middle is very vague.