Aside from taking a wrecking ball to Europe and fomenting an unspeakable Holocaust, the Nazis caused quite a bit of lasting damage to a great number of specimens of otherwise harmless cultural iconography.

The Swastika:

Up to the time of the Second World War, this simple asymmetric cross variation was indicative of nothing but East Asian mysticism, a Native American carpet weaving motif, and similar such things. And even in those uses, it was never a major symbol, being lumped in and under a dozen others employed by each such tradition. But now, it is a source of shock and confusion to discover the sign of the broken cross embedded in the decorations framing ancient temples in India.

The Toothbrush Mustache:

Oliver Hardy and Charlie Chaplin had them, those silly little horizontally centered half-mustaches. But thanks to Hitler, one can't wear one without evoking dictatorial megalomania. Interestingly, Hitler is hardly the only mustachioed menace, but nobody points and stares at those who wear mustaches like the bushy beast sported by Josef Stalin.

Goose-Stepping:

There is a scene in Indiana Jones and the Last Crusade wherein the elder Doctor Jones, taunted by a German officer, declares that "Goose-stepping morons like you should try reading books instead of burning them!" This is somewhat unfair to goose-stepping, as the person addressed was not a regular soldier, the type who would have paraded down city streets in shows of military strength. And so, he'd probably done little or no actual goose-stepping himself in his life.

Essentially, this exercise consists of nothing more then marching about via a procession of waist-high kicks done without bending the knee, forward motion accomplished by leaning slightly into the direction of the kick, and so having the foot meet the ground a bit ahead of where it started. It's actually kind of fun, and not a bad addition to a workout regimen -- but no longer, thanks to those Nazis so stridently using a march which had been practiced by other armies for centuries without raising an eyebrow.

The Roman Salute:

After two thousand years of essentially noncontroversial employment, the simple, if militaristic, form of salute embodied by a straight arm extended at a slightly upward angle, is no more. And, really, extracted from its ill reputation arising from Germany's adoption of it, it's a striking and powerful salute once thought fit to hail Caesar, the arm thrust out as if shooting from the heart itself. Nope, can't go around saluting people like that anymore (unless you're trying to imply something about their need to control others).

The Double Lightning Bolt:

The swastika wasn't the only symbol the Nazis took a proverbial hatchet to. The most famous insignia of the infamous Schutzstaffel (or, simply, SS) was a twin pair of lightning bolts, aligned side by side. But this symbol, like many others used by Nazi arms and organs, arose from harmless Germanic rune-craft. By the ill-starred fortuity of those bolts looking 'S'-like, they were taken up by Hitler's most rabid killers, leaving a taint upon them so bad that Rock band KISS can't even use their regular band logo in Germany, it's own S's typically being drawn with corners instead of curves.

(Clockmaker points out that "it isn't an »ill-starred fortuity of those bolts looking 'S'-like«; they're the runic letter S. (This is eliding certain alterations of Armanism and so on, but anyway.)")

The Word "Nazi":

Nazi, rhymes with Yahtzee. There are, it happens, other things historically named "Nazi." Sporadically, dispersed throughout the world, there have been towns named Nazi (apparently there are a bunch of them in present day Iran, though in this current climate of warmongering and weapon seeking, that may not be considered laudatory). There have been people in times of yore, in myth and legend, for whom Nazi was a given name or a last name (in the Nineteenth Century, even, a semi-popular girl's name of the Indian subcontinent, pronounced 'nah-ZEE'). There was even once an ancient Sumerian god named Nazi-- since re-(ahem)-translated as Nanshe.

But no more. Those names are consigned to dust or rewritten as something phonetically similar but juuuust different enough to avoid depravation by association. And though time enough has passed that we might jokingly refer to a strict restaurateur as a "soup nazi," the word will likely take generations more to fall into other vocabulary uses without opening wounds and drawing approbation.

Arguments on Teh Internets:

Per Godwin's Law, every argument on the Internet will eventually devolve into Nazi analogies. But what if the Nazi's had never been? Who, then, would accusations and associations devolve into? Napoleon? Stalin, maybe? By presenting so horrific a plan for mankind, with such ripe symbolism for all manner of stereotyping the Nazis established an unmatchable comparison of organized evil such that any civil discussion can be stopped dead in its tracks by directing such an invective against one's opponent.