Because of the setting of Red Badge – one particular battle in the U.S. Civil War – the novel is easily pegged as both a war drama and historical fiction. (The characters are fictional; the setting and its details are not.) The novel is also famous as one of the great realist novels of its time.
Crane writes the story of this historic battle from the very real perspective of one young soldier, relying on experiences and objects rather than allegory or heavy symbolism. This perspective, as we talk more about in "Point of View," actually means the novel falls under a very particular subset of the genre called "psychological realism." The experiences presented are subjective rather than objective, which means they are real psychologically if not absolutely.
Another term you might hear thrown about in reference to Red Badge is "impressionism." This genre is very similar to psychological realism and refers to works that present things as they seem or as they are sensed, not as they actually appear. Think of a Monet painting; it’s not about what the water lilies look like, rather it’s about what the painter feels when he’s looking at them. The painter’s mood comes through to the viewer via the paintings. It’s the same thing in impressionist literature. In The Red Badge of Courage, we’re given accurate portrayals of Henry’s thoughts and inner conflict, which is ultimately more important to the story than the simple facts of the battle.