Alternate history is usually classified as a subgenre within the broader genre of speculative fiction. In its strictest sense, an alternate history story is set on our earth, with the same laws of causality and physics. No magic or fantastic elements intrude on the workings of our real world. No inappropriate technological advances rear their heads like wristwatches on the Roman soldiers of the movie Spartacus, or airplane shadows over the hills of Rhett Butler's Civil War plantation. To an alternate history purist, such as Harry Harrison, only one detail has changed. It may be a small one, like the nail for want of which a war was lost, or it might be an enormous one, like the popular trope "What if the Germans won World War II?"
Regardless of how big or small the departure from our timeline was, like the butterfly effect, it has world-changing, plot-driving consequences. It determines the boundaries and new configurations of our parallel world in the story. Harry Harrison's Stars and Stripes trilogy hinges on a letter not being rewritten, which eventually leads to an early end to the Civil War with a combined North and South fighting off a British invasion. Michael Chabon's The Yiddish Policemen's Union explores what might have happened if a Jewish homeland had been created in Alaska. Philip Roth's The Plot Against America is an alternate history that hinges on Charles Lindbergh defeating Roosevelt in the presidential election of 1940.
Taking a less stringent view of strict historical possibility and realism, there are hundreds of other stories that fall under the alternate history umbrella, yet make use of impossible or unrealistic plot devices. Do you want to know what would have happened if an alien invasion had curtailed World War II and the defeated aliens had stuck around to be a part of the '60s civil rights movement and counterculture? Check out Harry Turtledove's Worldwar and Colonization series. Is magic more your thing? Orson Scott Card's Alvin Maker novels of a frontier America where folk magic worked could be just the thing.
There are some types of stories that seem to be alternate history, but do not quite fit in to this model. A brief clarification is called for. Historical fantasy is much like alternate history in that it makes significant use of a historical time, place and culture. But in historical fantasy, the history is usually backdrop to what the story is really about: romance, sorcery, cryptids, and advanced technology. Advanced technology brings steampunk science fiction into the fray. Steampunk is a subgenre set most often in the Victorian era, but with the miracles of Jules Vernesian science fiction made real.
Future history attempts to predict what will happen. Most often they are written from a perspective hundreds of years in our future and depict the ebb and flow of an imagined history in the intervening years, often including outrageous new inventions, discoveries and social trends. The last category of fiction which tends to be a kissing cousin of alternate history are Time Travel stories. These tales depend on a character escaping her own place and time and the adventures that follow in those other times, with varying degrees of historical accuracy.
Alternate histories are the ultimate "what if?" Like any other type of speculative fiction, alternate history, when done well, forces a re-examination of our own place and time, while scratching our natural human itch for adventure, exploration and discovery. Alternate history is perhaps the best suited of the subgenres for showing us the wonder of the world around us, and how it came to be the way it is.