Hi, I can give a brief synopsis although, to be honest, it's perplexing even for myself, although I am only a second year. I'll offer my understanding thus far, and am open to correction by those with more expertise than myself.
As John Gregson remarks, "of all the ‘classical’ ideologies, socialism is perhaps the most complex." The first development we must make in this answer is that socialism is more accurately termed 'socialisms' (plural) due to the splintering of various sub-variations of socialism into ideological components which in some respects directly contradict and reject one another. In order to understand these schisms, we must begin by observing the history of socialist thought. Contrary to popular belief, Karl Marx was not the father of socialism, but rather, the father of scientific socialism. There were a number of intellectuals who are referred to as pre-Marxian socialists (Marxian being someone who advocates for Marx's economic theories, while Marxist is someone who advocates for his theories as a whole) who have been labelled utopian by Marx, i.e. what they desire is not achievable or tangible for whatever reason. In the case of the pre-Marxian socialists, they did not observe analytically the development of history in a chronological manner, as did Marx. The three intellectuals credited with establishing socialism are Claude-Henri Sain-Simon, Robert Owen, and Charles Fourier. Marx is exalted over these thinkers due to his creation of what has been termed by him as scientific socialism. He interpreted history as the history of class struggle. This is also referred to as historical materialism. Essentially, since society evolved and formed, consistently there has been a pattern of one class benefitting off of the struggles of a lower class. Marx and Friedrich Engels wrote in the Communist Manifesto "Freeman and slave, patrician and plebeian, lord and serf, guild-master and journeyman, in a word, oppressor and oppressed, stood in constant opposition to one another..." Thus he is viewed as the father of socialism.
On to the differences in socialisms. The Bolsheviks and V.I. Lenin revised and adapted Marx's theories to suit their vision and their circumstance in Russia. After their revolution in 1917, the world witnessed a state which would be regarded as "communist" by the general population, particularly in the United States. This, paired with the failed German Revolution of 1918-1919, led to the division of socialism into two camps: communism and social democracy. Social democracy can be, and is, referred to more so as revised classical Marxism, tailored to contemporary society and thus prevailing over its predecessor. The general consensus regarding the difference between socialism and communism is boiled down to 'means and ends'. Here, the two ideologies differ. Rather than paraphrasing, I will paste Paul Wetherly's writings on communism and socialism below for a comprehensive understanding of their differences.
Communism can be understood as part of the tradition of revolutionary socialism and the most significant starting point can be traced back to Marx and Engels’ Communist Manifesto. These ideas have come to be known as Marxism and are so closely linked to the concept of communism that some have argued that Marxism is the theory and communism is the practice (Eatwell and Wright 1999: 104, 116)...
Communism, as specified by Wetherly, is, in essence, the practice of revolutionary socialism/Marxism. In this light, we can conclude that communists reject the state due to Marx's perspective on the state. The state, according to Marx, is an agent of the bourgeoisie in oppressing the proletariat. Thus the de-establishment of the state is sought out by true adherents of Marxism. Of course, what we witnessed in the twentieth-century "communist" states was a state possessing over-arching powers throughout its population, and quite bureaucratic and totalitarian in nature.
Wetherly further specifies:
Since the beginnings of socialism, the question that has separated socialists is how best to achieve its aims. Should socialism be brought about through gradual reform and working within the confines of the current parliamentary system? Or can only a radical overhaul of society bring about any genuine social change? It is the latter position that distinguishes communism from other forms of socialism. Social democrats tend to accept that it is both desirable and realistic to aim to implement socialist values within the confines of the capitalist system. Communism distinguishes itself from other forms of socialism through the way it views society as being irreconcilably in conflict between (p. 111) the two main classes of capitalism. This means that communists tend to believe that socialism cannot be implemented peaceably, or through reform, as the capitalist class will necessarily oppose the implementation of a system that runs directly counter to its own interests (see Case Study 4.1). In order to bring about real, meaningful social change, a working class revolution was necessary that would suppress the capitalist class and bring about a new phase of human society—communism.
I hope this answers your question - I was in a rush to finish so I may return to edit the answer, but if you have any questions feel free to ask me.
Edit: Quote blocks.
Sources Consulted:
Paul Wetherly, Political Ideologies, Oxford University Press.
Peter Ferdinand, Robert Garner, and Stephanie Lawson, Politics, Oxford University Press.
Karl Marx and Friedrich Engels, The Communist Manifesto, Oxford University Press.