Well, yesterday was Germany's day to celebrate and today it's their western neighbor's turn. While the Francophone community on io9 is no doubt out partying, I'll do my best to inform the rest of us Germanic-speaking barbarians what the big fuss is.
Bastille Day is a national holiday in France commemorating the start of the (First) French Revolution. Specifically, it marks the day on which a group of rebels seized the Bastille, a legendary fortress-prison located in Paris. It is, to make a bit of a rough comparison, the equivalent of Independence Day / the 4th of July in the United States, marking the end of the monarchy and the beginning of a new era of liberty, equality, and fraternity.
Well... sort of. Like many (even most) revolutions, the French Revolution was an infamously messy affair and it didn't really usher in the kind of social progress a lot of its early leaders were hoping for. King Louis XVI and his family's deaths were followed swiftly by the Reign of Terror, then the Thermidorian Reaction, then the dictatorship of Napoleon, and then finally the restoration of the Bourbons themselves. Democracy never really entered into the equation, at least not for any lengthy period.
That being said, the French Revolution (and by extension Bastille Day) still marks an important landmark in not only French history but European history as a whole. Although the French Revolution ultimately failed to establish a (lasting) democratic republic in France, it did successively alter the relationship between the French people and their monarch. When the Bourbons were restored at the end of Napoleon's reign, it was as a constitutional monarchy, fashioned in part after the British model, rather than the absolutist system which had reigned before. And the Napoleonic Code - one of the first examples of a civil law code which made no legal distinction between individuals on the basis of birth and which promoted the freedom of religion - would eventually become the foundation of European law as a whole.
In short, the French Revolution marks the widespread distribution and flowering of republican constitutionalism in Europe and classical liberalism. Many of these concepts had existed prior in isolated cases such as Great Britain or the Netherlands, but it was the Revolution (and the subsequent wars) which led to their proliferation across the continent. As such, while the human cost of the French Revolution or other uprisings like it should never be forgotten or trivialized, it's also important to recognize the legacy they leave behind in our laws, our institutions, and even our principles.
So celebrate without guilt, French-speaking niners, even if Edmund Burke disapproves. Besides, it's not as if the 4th of July is without its critics (or Anglo-American history without its legacy of bloodshed in the pursuit of political goals).