Without some form of conflict, no matter how subtle or minor, there is no story and, thus, no great literature. Although many classics don't feature much action or overt conflict, they still manage to give great significance to understated conflicts, as in in Virginia Woolf's novel "Mrs. Dalloway." Conflict builds a story and hold's the reader's attention.
Sometimes it's fairly easy to identify conflict in a literary work, but that's not always the case. If you're reading a novel about the Civil War, the conflict may be the large-scale struggle between the Union and the Confederacy, or it may be a single character's internal struggle for personal honor--even if treated ironically--as in Stephen Crane's short classic "The Red Badge of Courage." To determine the nature of the conflict, look at the main characters and discover what they want from themselves, from the world, or from each other. Once you understand the characters' relationships and motivations, you'll understand the conflict (or conflicts).
When characters are battling with themselves, you have internal conflict. Internal conflict consists of a character dealing with personal demons. This doesn't always mean the character is crazy or emotional. There are several types of internal conflict. Characters can struggle with their past, their heritage, their morals, their ideals, their desires, and much more. Classic novelists such as Henry James, Marcel Proust, James Joyce, and Virginia Woolf have depicted internal conflicts in extremely complex and subtle ways.
External conflict is usually a bit easier to see in literature. Unlike internal conflict, external conflict consists of two or more characters or objects in battle. External conflict does not have to involve swords and guns. Instead, it can be Elizabeth Bowen in conflict with Mr. D'Arcy in Jane Austen's classic study of manners and misunderstandings, "Pride and Prejudice." Some common types of external conflict pit protagonists against other people, society at large, or nature (as in Jack London's story "To Build a Fire"). Of course, there are many possible variations and approaches.
The satisfying part about reading fiction with good conflict is how it all turns out. Does Sherlock Holmes solve the mystery? Does Beowulf slay the dragon and save the kingdom? Does the narrator of "The Aspern Papers" obtain the papers he covets? Good conflict ends with what's known as the resolution of the narrative. Traditionally, stories resolve conflict in some way although modern writers such as Anton Chekhov, James Joyce, and Italo Calvino sometimes leave the reader hanging by introducing more conflict or leaving some mystery about whether or not the conflict is resolved.