If the Paris Commune teaches us one lesson, it is that insurrection is the most sacred of rights and that what is required in politics is, as Danton said, "De l'audace, encore de l'audace, toujours de l'audace"
The Paris Commune clearly demonstrated that it is only the working class that resolutely stands up to defend liberty, democracy and the rights of all peoples and against imperialist enemies. The Commune demonstrated that when workers are in power we get the most advanced, revolutionary and progressive ideas being implemented because the working class is the only progressive force in capitalist society.
In terms of democracy, the structure the communards implemented was the most democratic seen in contemporary European history and challenged the very nature of bourgeois democracy. Of course, the Commune was anathema to the French and Prussian bourgeoisie and the subsequent actions in crushing it showed the brutality which bourgeois society descends to when threatened. The opposing forces had to fight street by street, metre by metre, to conquer Paris in the teeth of the most ferocious workers’ resistance, itself a testimony to the popular support for the Commune.
The defeat of the revolution was only possible due the collaboration of the reactionary Versailles regime with the help of imperialist Prussia. The present Anglo-American led imperialism of the western ruling elite now stands in the same relation to the world, as the previous European decadent reactionary forces were to Europe.
The Paris Commune deserves to be remembered not only for its glorious attempt to implement the only true democratic structure seen in western Europe but also for the sheer brutality of the bourgeois response.
For me it's the amazing flowering of political awareness and expression that's most inspiring about the Commune.