The terms "communism" and "socialism" were, when they first became current in the labour movement and left in the 19th century, pretty much signified the same thing, but with variable shades of nuance. Since then they have, at various times, acquired different meanings.
Generally AWL prefers to use the word "socialism", because it is widely understood that there are many versions of "socialism". We then further explain that our idea is working-class socialism, which implies also that it is democratic socialism and revolutionary socialism.
We consider ourselves "communists" in the sense in which Marx, Engels, Lenin, Luxemburg, Trotsky and others called themselves "communist". Sometimes, however, "communism" is taken to mean a social system modelled on Stalin's USSR, or a political current trying to bring about a social system of that sort. We are definitely not "communists" in that sense.
Socialism is the older word. The Merriam-Webster dictionary dates "socialism" as first used in 1833, "communism" as first used in 1840. Whether they preferred one word or the other, or used both indifferently, communists and socialists generally reckoned that both indicated a society based on sharing and equality, and that this idea (in varying forms) was older than either word.
In 1848 Marx and Engels produced a pamphlet entitled "Manifesto of the Communist Party", generally known since then as "The Communist Manifesto". It was written for a political group they had joined just the previous year, called the "Communist League", which in turn had been formed by the merger of the "League of the Just" and the "Communist Correspondence Committee".
In 1872 Engels wrote a preface for a new edition of the Communist Manifesto, and declared that it was "the most widespread, the most international production of all socialist literature, the common platform acknowledged by millions of working men from Siberia to California" (emphasis added).
Engels continued: "Yet, when it was written, we could not have called it a socialist manifesto. By Socialists, in 1847, were understood, on the one hand the adherents of the various Utopian systems: Owenites in England, Fourierists in France, both of them already reduced to the position of mere sects, and gradually dying out; on the other hand, the most multifarious social quacks who, by all manner of tinkering, professed to redress, without any danger to capital and profit, all sorts of social grievances, in both cases men outside the working-class movement, and looking rather to the 'educated' classes for support.
"Whatever portion of the working class had become convinced of the insufficiency of mere political revolutions, and had proclaimed the necessity of total social change, called itself Communist. It was a crude, rough-hewn, purely instinctive sort of communism; still, it touched the cardinal point and was powerful enough amongst the working class to produce the Utopian communism of Cabet in France, and of Weitling in Germany.
"Thus, in 1847, socialism was a middle-class movement, communism a working-class movement. Socialism was, on the Continent at least, 'respectable'; communism was the very opposite. And as our notion, from the very beginning, was that 'the emancipation of the workers must be the act of the working class itself', there could be no doubt as to which of the two names we must take. Moreover, we have, ever since, been far from repudiating it".
From the 1880s a third term was used: "Social-Democrat". Today that term is usually taken to describe people who want some social-welfare reforms added to capitalism, rather than people who want a whole new society based on solidarity and equality. Until World War One, however, it was used very differently. It denoted those who agreed with Marx and Engels in seeking to achieve "socialism" or "communism" by working-class struggle and by qualitatively expanding democracy.
World War One and then the Russian workers' revolution of October 1917 brought sharp divisions among those who had called themselves "Social-Democrats".
In April 1917 Lenin proposed to the group he was a member of - then called the Russian Social Democratic Labour Party (Bolshevik) - that it change its name to "Communist Party".
"We must call ourselves the Communist Party — just as Marx and Engels called themselves.
"We must repeat that we are Marxists and that we take as our basis the Communist Manifesto, which has been distorted and betrayed by the Social-Democrats on its two main points: (1) the working men have no country: 'defence of the fatherland' in an imperialist war is a betrayal of socialism; and (2) the Marxist doctrine of the state has been distorted by the Second International".
His proposal was accepted. The Bolsheviks came to call themselves "Communist Party", and so did the groups which came together in other countries in the years after 1917 to try to bring about workers' revolutions like the Russian one.
Over the 1920s, bureaucrats led by Stalin made a counter-revolution in Russia, but were able to keep the name "Communist" for themselves and for the much-degenerated groups which they transformed the once-revolutionary Communist Parties into worldwide.
The activists who had supported, fought in, or led the October 1917 revolution, and now made a rearguard fight against the bureaucrats, used various different names at different times: Left Opposition, communist-internationalist, socialist, socialist-worker, Bolshevik-Leninist, Fourth International... Mostly they became known by the name the Stalinists gave them: Trotskyists.
For varying periods in some countries they tried to contest the title to the label "communist" with the parties aligned with the Russian government (and, later, those aligned with the Chinese government).
It is rare now for "Trotskyist" (working-class socialist, revolutionary democratic socialist) groups to name themselves "communist", if only because that name will connect them in most people's minds to the Stalinist states and movements, and maybe deter people who would sympathise with the groups' real aims (working-class democratic socialism) from inquiring further.
We chose the name "Alliance for Workers' Liberty" for ourselves, rather than re-use the terms "communism" or "socialism" (which, after all these years, always need explanation about which sense, out of many, you're using those terms in), to convey that our aim is the liberation of the working class through the activity of the working class.