The narrator of "The Most Dangerous Game" is a third person narrator. The narrator is also a third person limited narrator, because he doesn't know and explain everything to the reader. Zaroff's background, thoughts, and motivations are not explained to the reader unless Zaroff himself is speaking. The reader only knows what Rainsford is thinking, which is why it is a limited narration.
The conflict is mainly a person vs person conflict. Rainsford and Zaroff are in a battle to the death against each other. It's both a physical battle and a battle of wits. There is some person vs. nature conflict in the story too. The island is dangerous. Rainsford doesn't know all of the ins and outs of the island, so the environment itself is a source of conflict. There is a bit of person vs. self present in the story too. That's evident when Rainsford is wrestling with his own thoughts and feelings about killing another man.
Part of your question asked about the story's plot. The Enotes summary is linked below, but I can give you a simple plot chart overview as well.
Exposition: Rainsford is on the boat, and the reader comes to learn that he is a great hunter.
Rising actions: Rainsford falling overboard. Rainsford arguing with Zaroff about human hunting. Rainsford surviving 2 days as Zaroff's prey.
Climax: Rainsford escaping by jumping off of a cliff into the ocean.
Falling action: Rainsford sneaking back into Zaroff's house and killing him
Conclusion: Rainsford sleeping peacefully in Zaroff's bed.