Here's the Cliff Notes version as I understand it:
In early 1871, France was at a low point, having just lost the Franco-Prussian War. Paris had particularly suffered in this conflict; during the months long siege by the Prussians, there were severe food shortages and people were reduced to slaughtering pets and vermin to stay alive. When the Germans finally left, the city was in tatters, social and economic conditions at a low point. Discontent among the working class, that had been simmering for a decades, came to a head.
The future of Paris, however, lay not in their hands but with the national government which, like the kings before them, feared what Parisians might do if given the right to control their own city. And so the workers took matters into their own hands, forming a new government called the Commune and immediately issued various decrees issued to improve quality of life: educational reforms, separation of church and state, relief of debt for working men, and the abolition of night work. The national government and its army retreated to Versailles.
And then from what I can gather, chaos ensued. The revolutionaries were not fully prepared for the task of governing and the military powers were hell bent on breaking the revolutionaries at any cost. No one was willing to negotiate. Blood ran in the streets and fires burned, destroying many parts of the city including the Tuileries. At the end, the national government was triumphant, the revolutionaries shot or exiled. And nothing much changed.
And yet the fight continues. A conversation with friends who live in the 7th and 8th criticized a recent exhibit on the Commune at the Hotel de Ville as being predictably skewed towards the revolutionaries, the mayor of Paris being after all a Socialist. In the 20th, the mairie sanctioned creation of a mural celebrating the anniversary. (To fully appreciate this mural, click on the picture below and it will enlarge.)
And opposite this wall, a billboard calls upon local residents to join the spirit of the Commune in carrying out unfulfilled promises for social and economic improvement.
Vive la Commune? It's not for me to say. But as for keeping history alive and using it as a tool for social progress, I'm all for it.