Though I'm not 100% sure of your question, I'm going to go ahead and assume you're referring to the "feud" or conflict between Rainsford and General Zaroff. The feud began because General Zaroff had heard that Rainsford was the best hunter in the world. Zaroff then intentionally caused Rainsford's ship to crash on his island so that Rainsford would be forced to participate in the game that Zaroff created.
The game, or the escalation of the conflict is that Zaroff wanted to hunt Rainsford himself. Zaroff tells him that if Rainsford could stay alive and evade capture for a certain amount of time then Zaroff would let him go. However, the conflict resolves differently than Zaroff expected. Rainsford not only managed to evade capture, but ends up killing Zaroff in the end.