Most of these terms (exposition, rising action, climax, falling action, and resolution) are the general sections of any story. They help you organize a story in your mind and visualize it as a structured set of events.
Although I'll say that, for example, the exposition of The Cay is Chapters 1-2, and that the rising action is Chapters 3-11 and so on, other readers might have slightly different ways to make those divisions. But the general idea is that these terms help us make a map of the plot in order to understand it.
First, the exposition of any story is the very beginning, where we learn what the setting is, who the characters are, and what they basically want.
So, you can find the exposition of The Cay in Chapters 1 and 2, when Phillip and his parents are wondering whether they should try to leave their Dutch island of Curacao to head to Virginia, where it might be safer. We learn that World War II is going on, and that there are shortages of food and water where Phillip is.
After the exposition comes the rising action, when, basically, action starts happening!
The rising action of The Cay comprises Chapters 3 through 11, when Phillip and his mom board the ship to head for Virginia, a torpedo causes their ship to sink, and Phillip ends up on a raft with an old black man named Timothy. They float in the middle of the ocean, hopelessly, until they finally reach a tiny deserted island, and they busy themselves with making a temporary camp there. They build a signal fire so that they might get rescued, and after some tense conversations, Phillip realizes that he respects Timothy and wants to be his friend. Timothy wonders if they are cursed with bad luck.
After the rising action comes the climax: the most exciting part, or the most difficult experience that the main character must endure. Some stories have more than one climax, and different readers could disagree about which one is "the" climax or the most important one.
In The Cay, I'll say that the first climax is in Chapter 12, when Timothy becomes extremely sick with malaria. That's when readers realize that Phillip will be totally alone and helpless if Timothy dies, and it's when Phillip really has to struggle hard to learn to feed himself and keep himself safe from hurricanes.
The second climax is in Chapter 15, when the hurricane hits and both characters have their strength tested by wind and waves. Because this event is so dramatic and chaotic, with so many preparations beforehand and so many repercussions afterward, it's probably the climax that most readers would identify as the more important one.
Falling action is all the events that occur after the climax(es). For The Cay, the falling action is Timothy's death at the end of Chapter 15, followed by Phillip's mournful thoughts and continual, solitary work to keep himself fed and alive on the cay, alone, throughout Chapters 16-18.
And lastly, the resolution is how the story ends. In other words, how is the conflict resolved, or how does the struggle end? Who wins?
In The Cay, the resolution takes place in Chapter 19, when Phillip is finally rescued from the cay, and he is united with his parents.
Let's consider the conflict separately from all the previous elements. Conflict is present in any of those sections of a story, but most notably the rising action and the climax. Conflict is when characters struggle: when they have experiences that are difficult or that require them to adapt and learn. You can specify what kind of conflict is going on by saying who or what the character is struggling against.
In The Cay, the conflict is between:
Phillip and his own misunderstanding of and racism toward black peoplePhillip and his own new blindness, including his struggle to be self-sufficientPhillip and Timothy, as the boy realizes that the old man is not his enemy but his friend and his only hope for survivalTimothy and his malariaThe two people and nature: the conditions on the cay, especially the hurricane, present major difficulties
