Anarchism used to be a huge movement that inspired dozens of acts of international terrorism, from the killing of an American president and multiple European monarchs to the 1920 Wall Street bombing.
Where did that rash of attacks come from, and where did it go? Why don't we fear bomb-throwing anarchists today?
A tactic that has been to central to anarchism since the 1870s is that of "propaganda of the deed", The idea is that by doing some kind of political action, you can inspire others to act in a similar way and lead towards revolution. While this is usually associated with violence, it also has nonviolent applications such as the group Food Not Bombs who give free food away that would otherwise have been wasted. Combine this with the anarchist attitudes that "peace without justice is not peace at all" and a rejection of the legitimate monopoly on violence, and it becomes very easy to justify violent actions in order to make a revolution. While some major anarchists supported this, it's worth noting that the majority believed that these assassinations, whether morally acceptable or not, would only bring further political persecution.
And further political persecution did come. Followers of the "insurrectionary" anarchist Luigi Galleani sent out 44 bombs in April-June of 1919 killing one city watchman and one bomber. This lead to hundreds of deportations and countless searches and arrests. Most historians believe the decline of the anarchist movement began in 1920 after the trial of Sacco and Vanzetti, a shoemaker and a fish seller respectively who were charged with armed robbery. Both were anarchists who advocated revolutionary violence and while Sacco might have been guilty, Vanzetti was clearly innocent. Both were executed by electric chair in a trial with a large amount of media coverage.
At the same time, you had a rise of syndicalism, a trade union movement that aims to form "one big union" that organizes all workers of the world, and then through a general strike overthrow the capitalist system and have workers occupy their workplaces. One such union, the Industrial Workers of the World or the "Wobblies", had over 150,000 members in 1917. Most were rural workers in the US, especially immigrants. Syndicalism provided a way for workers to overthrow the government without propaganda of the deed assassinations, so radicals could organize for this instead of take their frustrations out through violence. But syndicalism was not too last as political and media pressure picked up during the First Red Scare of 1919-1920 and IWW membership began to decline. In 1924 the party had a split from which they never truly recovered.
Sources:
Eric Thomas Chester, The Wobblies in Their Heyday: The Rise and Destruction of the Industrial Workers of the World during the World War I Era
Peter Marshall, Demanding the Impossible: a History of Anarchism
Mods let me know if my citations or anything else is unsatisfactory, I'm not actually a historian I'm just a bored anarchist with too much free time.
Edit: I want to add a quote from David Graeber, a well known contemporary anarchist and anthropologist, to give people an idea of attitudes at the time among anarchists towards the assassinations.
There was a spurt, towards the beginning, of calls for the assassination of heads of state, it was quite brief and anarchist spokesmen and organized groups quickly withdrew support from this strategy as counterproductive. Nonetheless, following decades saw a continual stream of dramatic assassinations by people calling themselves anarchists. I am not aware of any actual assassin during this particular period who actually was a product of these anarchist organizations, much less were their actions planned or sponsored by them; rather they almost invariably turned out to be isolated individuals with no more ongoing ties to anarchist life than the Unabomber, and usually about a roughly equivalent hold on sanity. It was rather as if the existence of anarchism gave lone gunmen something to call themselves.
Taken from page 212 of Direct Action: an Ethnography
There are a few previous threads with more or less the same question, in case nobody comes in here to answer you: https://www.reddit.com/r/AskHistorians/comments/5z5cpx/why_was_anarchism_a_thing_around_the_findesi%C3%A8cle/
You can also visit /r/anarchy101 to learn more specifically about anarchism, if that helps.
It's worth pointing out that responses and attitudes to violence in anarchism have always varied. Some anarchists have been pacifists, others have believed that some variety of response to state violence is necessary, usually with an emphasis on self-defense with a revolutionary character.
However, around the turn of the century some people on the left, and not just anarchists, believed in the idea of "Propaganda by the deed", allied to the anarchist emphasis on direct action. The idea was supposed to be that such direct acts would incite the workers to greater awareness of the class conflict inherent in the system.
So it's more accurate to say that these assassinations and bombings were more of a passing phenomenon within anarchism, which around the time of the First World War went out of fashion, and were always regarded with equivocation even within the anarchist movement. If you read modern anarchist writers, they are usually uncomfortable or skeptical about the terroristic phase in anarchism.
Even in the case of individual anarchists you can find that moral ambiguity about the use of violence. Leon Czolgosz, for instance, who shot President McKinley, was supposed to have been inspired by Emma Goldman, but she herself did not condone the action directly. Such acts in anarchist literature were usually seen as desperate actions by poor workers driven to extremism, suffering from idealism, and therefore a further indictment of the system. Goldman's partner Alexander Berkman also spent a long time in jail for a futile and ineffective attempt on the life of a business magnate. So that may have given her more insight into the limited effectiveness of the tactic of the "Propaganda by Deed". Czolgosz himself seems to have been a neurotic individual who was probably acting out his personal problems in dramatic fashion.
Other supposedly anarchist terrorists were definitely frowned upon by some factions of anarchists, while celebrated by others, examples being Ravachol and Nechayev. Nechayev and some of the Russian anarchists in their philosophy cross the line into nihilism, sometimes self-consciously, sometimes not.
After the First World War and the October Revolution in Russia, other tactics were emphasized, and anarcho-syndicalism became popular, being especially strong in France. The emphasis on direct action continued, but the idea of direct action was more focused on collective action by syndicats (or labour unions) in such actions as strikes and occupations.
Anarchism itself as a wide-ranging tendency has never really gone away though. If it has at various historical moments seemed subdued or gone into a fallow period, it is most likely because of strong state repression, or because other forms of left organisation (Russian-style communism, Trotskyism, vanguardism, Marxism, Maoism, the various types of more authoritarian communism) predominated and became fashionable.
It's also true that the assassination of European monarchs was not always inspired by anarchism, some of the assassins being motivated by other political ideals which involved anti-monarchism, such as republicanism. Lots of people in history have been assassins, for all sorts of reasons, sometimes for political reasons but sometimes for merely personal grievances.
So to answer the question, "Where did that rash of attacks come from", it came in the case of the anarchists from the notion of "Propaganda by the deed", which was the somewhat romantic idea that by targeting selected state officials and other representatives of the state, seen as agents of tyranny, the people might be inspired to revolutionary action thereby, or at least that the conflict inherent in the system might be highlighted. And, "where did it go". It gradually fell out of fashion like other features of the period like Nietzscheanism and Social Darwinism, and by the 1930's a popular form of anarchism was revolutionary syndicalism, as exemplified in the anarchist movement in Spain which found popular expression during the Spanish Revolution and Civil War which began in 1936.
Direct action tactics experienced a revival of sorts in the 1960's and 70's with various groups of the left in Europe and America. Examples include the Angry Brigade in Britain, the Baader-Meinhof gang in Germany, the various groups in Italy such as the Red Brigades (responsible for the assassination of Italian statesman Aldo Moro), and in America, the Weather Underground. Some of these were only tangentially associated with anarchism though, being more inspired by the Maoism and vanguardism fashionable at the time.
Hope this is detailed enough.
I'd like to add another element to this discussion: some political scientists would argue that an additional factor was a change in government intervention in industrial conditions.
Between the end of the American Civil War and the beginning of the Great Depression, the American economy was rapidly industrializing, yet the politics of the day failed to reconcile the effects of a new means of production with the old Jacksonian ideal of the government as a 'night watchman' over a free-contract economy. Non-intervention and the freedom of contract works relatively well (at least on a broad scale) when you have small owners making one-to-one contracts, and either party can walk away, but industrialization created a power imbalance where workers couldn't walk away and continue to feed their families. People work suffering under shockingly bad working conditions, and the state was primarily using its monopoly on violence to prevent them from organizing, and rarely using it to improve working conditions. All branches of government on every level in America were somewhat dedicated to this ideal. It is unsurprising that many workers felt that the only way they would be protected was through violent overthrow of the government.
That all changed when Roosevelt and the New Deal Democrats rolled into power (the new understanding actually started earlier in the progressive era, thanks in no small part to pressure from radicals, but the transition completed in '32). A new constituonal and legal understanding took hold in which the government could and should intervene and regulate to protect workers. The New Deal Coalition included the unions at the table and, in doing so, deradiclaized many of them as well. The New Deal and Great Society coalitions presided over a time of unprecedented equitable economic growth, so few individuals were unhappy enough with the political economy to agitate for violent overthrow of the government.
Source: Gillman, Graber and Williams, American Constitutionalism, Vol 1 & 2
because propaganda by the deed didn't get results. why keep doing the same thing if it isn't working?