I'm trying to figure out: how relevant was anarchism as a political movement? (I'm thinking primarily of the period of time between when Proudhon was writing in the mid-19th century and the Russian Revolution in 1917.) Was it ever seen by large numbers of people as an alternative to whatever the current state of politics was at the time?
Here's the start of an answer: "Anarchism" was not a widely used keyword until 1880 or so, around the time of the establishment of the The International Working People's Association or "Black International." Prior to that, anarchist ideas played a role in organizations like the International Association of the 1850s and the First International, and informed some elements in uprisings like the June Days of 1848 and the Paris Commune. In North America, they were also part of the movement for "equitable commerce," informed the radical wing of the abolitionist movement and inspired the organizers of the various New England reform leagues. But they were, in this early period, generally expressed in the context of larger socialist and/or internationalist movements. Those movements were marked by all sorts of internal struggles, including significant disagreements between anarchistic factions. It was arguably not until after the splits in the International that the various elements, anarchists among them, would emerge as movements in their own right, with the new divisions drawing anarchists together as socialism and the international labor movement split apart.