When you, the reader, sit down to analyze a book, you must remember that whatever you choose to say is subjective. It's your opinion and you're entitled to it. Even if it means trashing a widely acknowledged author.
To give everyone the courage to speak their mind, the Facebook page Haters of Goodreads are sharing some of the funniest reviews that have appeared on the literary website.
Calling The Catcher in the Rye "the most overrated 'classic' of all time", refusing to finish Swann's Way due to Proust "discussing the smell of his chamber pot after having eaten asparagus"... It's all there!
If you, however, want to do (and write!) more critical analysis of the books you read, the University Writing Center at Texas A&M University suggests to begin by summarizing the basic plot — this will help ground you in the story.
Then, research the author's background and other work. This can give insight into their perspective and bias, as well as reveal what they might be commenting on. As an example, the University Writing Center mentions Tolkien's The Lord of the Rings. It's about a group of friends who embark on an epic journey and fight a great war. But knowing Tolkien fought in the Battle of Somme during World War I and that his closest friends were killed helps explain his sentiments about war.
Other questions about context can stem from the story itself: think about the narrator's personality and their role in the story. Also, it can be a good idea to consider who the narrator is addressing.
Aristotle, Nichomachean Ethics
Next, we have setting. When and where a story takes place can be of huge importance. Where the author's story is placed? Why the author made that decision?
Many stories would be irretrievably altered if their setting were different and setting is, therefore, vital for interpreting the story's meaning. To illustrate this point, the University Writing Center highlights the setting for Faulkner's work — the American South after the Civil War. It is essential to his overall message. Faulkner's characters are people who can't move on, and through them the author suggests that the South similarly can't get past the Civil War and the wrongs of slavery.
By the way, storylines usually evolve in patterns, so identifying essential plot points might help you to analyze, interpret, and explain the story as well.
Rousseau, The Social Contract
But characters are the driving force behind stories, both the major and the minor ones. Like the above-mentioned Faulkner example suggests, authors can use them to broadcast their most important messages. You won't be able to analyze every character in a book, but pick out several important ones to consider.
For this, you can use the following questions: What are the character's main personality traits and why did the author give him these traits? What is the character's role in the story? What are the character's morals or ethics? Why does the author give him those? Why does the character do what he does? Why did the author make him act that way? What is the character's relationship to other characters and why?
Also, various literary devices help convey meaning or create a mood. Look for allusions, irony, symbolism, and other "tools" in a story to identify key points and their contribution to the author's overall message.
After you've worked on the story for so long, you should start to get a sense of its major themes, the big ideas that authors comment on throughout the work. Common themes are good vs. evil, human nature, religion, social structure, authority, coming-of-age, human rights, and so on. Books typically deal with multiple themes, some more obvious than others.
Once you complete the analysis, develop a thesis that makes an arguable claim about the text — like "wtf?" — and post it on Goodreads.