The most obvious type of conflict in the story is the conflict of man vs man. Rainsford is in a fight for his life against Zaroff. The two men are squaring off and battling each other's wits and skills at hunting.
A second type of conflict in the story is man vs. wild. This conflict affects both Rainsford and Zaroff. Both men are subject to the dangers that the island has, but Rainsford is at a disadvantage. He does not know the island and its dangers as well as Zaroff.
A third type of conflict is man vs. self. This conflict applies to Rainsford. As he is struggling to survive he has to reconcile his previous thoughts about hunted animals not experiencing fear. Very early in the story, Rainsford felt no pity for the things that he hunted. Now that he himself is being hunted, he has a different perspective on the matter. Also embedded in the man vs. self conflict is Rainsford willing himself to continue. He's in a pretty bad situation, and it takes a lot of self motivation to keep on fighting against a dominating opponent.
The word "game" has two meanings in the story. The first is that hunting is a game. It's something a person plays to win. Zaroff and Rainsford are playing a game to see who can win (survive). The other meaning is about how hunted animals are called "game." Rainsford is the game this time around instead of the person hunting the game.