While no formal definition exists, an alternative fact may be loosely defined as an "alternative interpretation" of a fact to suit one's one political or ideological purposes. Whether such an alternate interpretation is, or is not, valid typically depends upon the fact in question, and how it has been interpreted (or misinterpreted).
The phrase "alternative facts" gained traction in January, 2017, when presidential advisor Kellyanne Conway used it in a Meet the Press interview. The day after the interview, The Guardian described, "the introduction to common parlance of the term "alternative facts," while noting outcry that it is a fabricated term. Despite claims that it has a history of use as a legal term, it is not recognized as such in any formal sense. The Guardian further notes that searching law dictionaries did not yield any formal legal definitions of the term.
Conway later gave examples of alternative facts as being, "Partly cloudy, partly sunny" and "Glass half full, glass half empty," which are actually interpretations of the weather or the volume of water in a glass. An factual description of this volume would state there are, for example, 8oz of water in a 16oz glass, and would not include a subjective half full or half empty interpretation. (See objectivity vs. subjectivity under "What is a fact?" to the left).
Marine Le Pen, pictured to the left, is the current leader of Front National ("National Front") party in France. Generally regarded as France's most hyper-partisan political party, the Front National has a history of Anti-Semitism. As the the party's current leader, Le Pen has attempted to distance the party from this past, though one of her recent comments reported by the New York Times demonstrates that neither she nor the party has completely abandoned it.
In an interview on April 9, 2017, she referred to the Vel d'Hiv roundup (WSU login required), an incident on July 16 and 17, 1942, in which 12,884 Jews were arrested by French police officers under the direction of the Vichy government, the collaborationist French government that had acquiesced to the German occupation. In her interview, Le Pen stated that,
France wasn't responsible for the Vel d'Hiv. If there was responsibility, it is with those who were in power at the time, it is not with France.
This remark contradicts 22 years of official French policy. In a 1995 speech commemorating the Vel d'Hiv roundup, then-French President Jacques Chirac declared that the nation has a responsibility to accept its past actions as part of its history, including atrocities committed by its government.
Responding to the public outcry over her treatment of history, Le Pen tried to mount a defense by using logic that might be described as an alternative fact: she attempted to deny that the Vichy government was the government of France in 1942. She put out a press release in which she said
I consider that France and its Republic were in London during the occupation, and that the Vichy regime wasn't France.
The Times article describes this alternative interpretation of history as "historical revisionism," and states that, "the historians' verdict is different." It goes on to quote French historian Henry Rousso, who specializes in France's World War II-era history, regarding the French citizenry recognizing Vichy as their own government:
To isolate Vichy from the French population doesn't hold up for one second. You only have to look at the newsreels of the crowds applauding [Vichy leader Phillipe] Pétain.
The Times also notes some historically verifiable facts about the Vichy government: that it was recognized as France's ruling body by other world leaders and governments, many of which maintained embassies with it, and that many of its officials held elected office both before and after World War II -- time periods in which the French government did not maintain a policy of collaboration with any foreign government.
In short, it is a factual statement to say the Vichy government was the government of France in 1942. There is no such thing as an "alternative fact" that disproves or disputes this fact, nor is there any realistic room for an interpretation that places this fact in contention. Any attempt to claim there is falls apart on historical grounds.