...We celebrate it on the wrong day (there may be no single right one: the Declaration of Independence was adopted by the Second Continental Congress 2 July 1776 and signed, mostly, by 2 August) and often with an unclear understanding of what it meant, though most folks get the basics well enough: the birth of the United States. The glorious notion that claiming to have been handed a Crown by $DEITY or blood-right was nonsense, that men were clever enough to rule themselves and the sure knowledge -- stemming from a long period of benign neglect -- that the less obtrusive the government, the better off the people.
But though they were soon to be calling themselves "States," i.e, Nations, the former colonies were not all that "united" in the Summer of '76, acting instead in a loose confederation later codified by the Continental Congress passing the Articles of Confederation in November of 1777, Articles not ratified by a majority of the former colonies 'til March 1781. (It should be noted that the Second Continental Congress was the American Congress that began the custom, honored by all successor bodies, of kiting checks or equivalent paper. This provides yet another reason to keep government small).
The day our Founders told the Crown to take a hike is not the same thing as the day the United States became a nation: on that day, the 13 colonies became 13 nations. The "one nation, indivisible" guff took nearly a hundred years to gel and a far greater death-toll than than was paid in the War of Independence. (And even at that, it took a concerted PR pitch in the form of a nearly-compulsory oath to completely sell it -- see Francis Bellamy). What we share as U. S. citizens is more than a mere patch of land or a required oath to a symbol. We share a dream, a shared ideal, a solemn promise redeemed in blood. Don't lose sight of it in the celebration.
When was the most recent bit of revolutionary zeal exercised in the U.S.? Possibly 1961! (We'll ignore the cabals of secessionist plotters in Yreka and Key West and their occasional overt acts for now). The 13 colonies did it first and best.
On this day (or near enough) in 1776, brave, clear-thinking men put their fortunes, lives and sacred honor on the line for an idea: the wild, unproven notion that adult humans could govern themselves. 232 years later, it's worked out pretty well but our would-be Kings and petty nobles cry out more loudly every year that they know what's best for us and should dictate our lives and our fate. Green or to the Purple born, they're not to be trusted. Tell 'em what the Colonies told the King!