A future history is a postulated history of the future that some science fiction authors construct as a common background for fiction. Sometimes the author publishes a timeline of events in the history, while other times the reader can reconstruct the order of the stories from information provided therein.
A set of stories which share a backdrop but are not really concerned with the sequence of history in their universe are rarely considered future histories. For example, neither Lois McMaster Bujold's Vorkosigan Saga nor George R. R. Martin's 1970s short stories which share a backdrop are generally considered future histories. Standalone stories which trace an arc of history are rarely considered future histories. For example, Walter M. Miller Jr.'s A Canticle for Leibowitz is not generally considered a future history.
Earlier, some works were published which constituted "future history" in a more literal sense - i.e., stories or whole books purporting to be excerpts of a history book from the future and which are written in the form of a history book - i.e., having no personal protagonists but rather describing the development of nations and societies over decades and centuries. Such works include:
Asimov retained in part this earlier form of Future History in prefacing each of The Foundation Series stories by a pseudo-historical quotation from the "Encyclopedia Galactica". The same practice was followed by later writers, such as Jerry Pournelle in his own future history series.
Also Poul Anderson's story "Inside Straight" is prefaced by an excerpt from a future history book, with a very precise reference: "Simon Vardis, a short History of Pre-Commonwealth Politics, Reel I, Frame 617".
The essential difference is that the writer of alternate history is in possession of knowledge of the actual outcome of a certain event, and that knowledge influences also the description of the event's alternate outcome. The writer of future history does not have such knowledge, such works being based on speculations and predictions current at the time of writing - which often turn out to be wildly inaccurate.
For example, in 1933 H.G. Wells postulated in The Shape of Things to Come a Second World War in which Nazi Germany and Poland are evenly matched militarily, fighting an indecisive war over ten years; and Poul Anderson's early 1950s Psychotechnic League depicted a world undergoing a devastating nuclear war in 1958, yet by the early twenty-first century managing not only to rebuild the ruins on Earth but also engage in extensive space colonization of the Moon and several planets. A writer possessing knowledge of the actual swift collapse of Poland in World War II and the enormous actual costs of far less ambitious space programs in a far less devastated world would have been unlikely to postulate such outcomes. 2001: A Space Odyssey was set in the future and featured developments in space travel and habitation which have not occurred on the timescale postulated.
A problem with future history science fiction is that it will date and be overtaken by real historical events, for instance H. Beam Piper's future history, which included a nuclear war in 1973, and much of the future history of Star Trek. There are several ways this is dealt with.
First, some authors set their stories in an indefinite future, often in a society where the current calendar has been disrupted due to a societal collapse or undergone some form of distortion due to the impact of technology. Related to the first, some stories are set in the very remote future and only deal with the author's contemporary history in a sketchy fashion, if at all (e.g. the original Foundation Trilogy by Asimov). Another related case is where stories are set in the near future, but with an explicitly allohistorical past, as in Ken MacLeod's Engines of Light series.
In other cases, such as the Star Trek universe, the merging of the fictional history and the known history is done through extensive use of retroactive continuity. In yet other cases, such as the Doctor Who television series and the fiction based on it, much use is made of secret history, in which the events that take place are largely secret and not known to the general public.
As with Heinlein, some authors simply write a detailed future history and accept the fact that events will overtake it, making the sequence into a de facto alternate history.
Lastly, some writers formally transform their future histories into alternate history, once they had been overtaken by events. For example, Poul Anderson started The Psychotechnic League history in the early 1950s, assuming a nuclear war in 1958 - then a future date. When it was republished in the 1980s, a new foreword was added explaining how that history's timeline diverged from ours and led to war.