The word "buffalo" has a number of different meanings. First and foremost, it is used to describe various species from several different genera of large, bovine mammals, including Syncerus (including the African Cape buffalo), Bubalus (including the Asian water buffalo) and Bison (including the American buffalo, or bison).
By the way, all of those who claim that it is only proper to refer to the American version as "bison," you need to get off your "one true buffalo" high horse. Not only is buffalo an extremely common, generic term (the etymology traces back to Greek bous, "cow"), but it was the only English term used for more than a hundred years after the American animal was first discovered by Europeans.
"Buffalo" is an ancient, poetic word that flaunts modern conventions. While you sometimes see "buffaloes" as a plural form, it is most often used as a collective noun, so that its plural form matches the singular: "a herd of buffalo" not "a herd of buffaloes." This is important for the linguistic gymnastics it performs in my favorite sentence.
Buffalo is also a common place name; most notably it is the name of a large city on the shore of Lake Erie in upstate New York. And it can be used as an adjective to describe things associated with that city: "I hear Buffalo winters are brutal."
Finally, "buffalo" can be used as a transitive verb meaning "to intimidate by a display of power": "As mayor, I tried to buffalo the officer into not writing me a ticket, but it backfired and I ended up in jail."
Now a word about buffalo society. (Notice the lowercase b.) It is clear that the meaning of the verb is derived from the behavior of the animals. It is the nature of buffalo to buffalo; it's just what they do. A buffalo will buffalo anything that it can: people, bears, prairie dogs, you name it.
It should also come as no surprise that buffalo can and will buffalo other buffalo. And while buffalo herds do move around a bit, in general it is safe to say that Kansas buffalo will interact in this way with other buffalo from Kansas, as opposed to, say, buffalo from Wyoming.
And so here we come to our payoff, because if asked to describe the social interactions of buffalo in upstate New York (as I'm sure you will some day), the first thing that should now spring to mind is my favorite sentence of all time:
Buffalo buffalo buffalo Buffalo buffalo.
P.S. - You can also say "Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo." That's eight buffalo in a row, if you're counting. This version is grammatical in theory, but it really does stretch the rules of English grammar beyond all practicality. As a courtesy to the listener, the speaker of the eight-word version would almost have to insert some extra words to preserve the flow of the sentence: "Buffalo buffalo whom Buffalo buffalo buffalo in turn buffalo other Buffalo buffalo." Because of this, as opposed to the five-word version, I can't see myself ever using it.