Realism, Tragedy
The dog and the "dum-dum" die: yep, we'll go ahead and slap a "tragic" label on this.
Or we could get more official: tragedy usually features some main character who experiences a reversal of fortune from good to bad. This reversal is always brought about by an innate flaw of the character, or by a mistake that he or she makes. In this case, George's flaw is his trust in Lennie—a mistake that even he realizes by the end of the book. In the final section, George stands over Curley's wife's body and says, "I should of knew…I guess way back in my head I did" (5).
But that's where the "typical" part ends. Tragedies traditionally center on main characters who are big-shot-important-leader types, with steep falls from grace. But not George. George's fall from having a mentally handicapped friend and a dream to having no mentally handicapped friend and no dream is not particularly dramatic. But that's the way Steinbeck shows that all men matter: common men can also be the heroes of their own lives, and the victims of great tragedy.
Which brings us to realism, a type of writing that wants to convince you that everyday people matter.
Of Mice and Men employs a very particular type of realism called "naturalism," a type of literature where the narrator looks at the characters as though they're scientific specimens: objectively and dispassionately. (Check out "Narrative Technique" for more about our narrator as an observer). Naturalist novels often depict the world as a place where you have to fight to survive in a universe that has no morality and doesn't care about you.
Cheerful stuff, right? The strength of realism is that you don't need to be sold with emotional bells and whistles—the lives of people alone are enough to move you.