How does J. B. Priestley's "An Inspector Calls" represent socialism?

Socialism is presented by Priestley as more of a moral imperative than an economic rival to capitalism. In An Inspector Calls Priestley allegorizes what he sees as the morally destructive impact of unrestricted capitalism on people's lives. And that doesn't just mean those at the bottom of society like Eva Smith, but also the wealthy social elite such as the Birlings. Their souls have been completely corrupted by capitalism, and it takes the supposedly more humane spirit of socialism—as personified by Inspector Goole—to force the Birlings to face up to the reality of their moral bankruptcy.

Inspector Goole's lugubrious presence is intended to act as a reminder of the moral transformation that Priestley believes will come to Britain in due course. In advance of the impending arrival of socialism, those who have derived the most benefit from the capitalist system must examine their consciences and at least reflect on how they must change their ways in adapting to a new, more communitarian system.

Do you mean as an institution or as a philosophy?  If you mean the latter, the inspector himself represents socialism (or rather its values) by knocking on the Birlings' door and asking for a personal reckoning of each person's dealings with the suicide victim in question (a poor, bewildered, and abandoned pregnant girl who kills herself by swallowing poison). The moral "in a nutshell" is that as members of society, we are all implicated and we are all responsible for the down-and-outers living on the fringe of our material world.

Socialism as an institution is not really represented since it is the capitalistic system that the author is criticizing.

Priestley's silver spoon preoccupation with the plight of the poor are reminiscent of Charles Dickens,  Bernard Shaw and George Orwell (Eric Blair), all disgruntled with the inegality of wealth distribution and more particularly, with the flagrant disparity between word and deed of the social "do-gooders" of their times.

Check out the references below for further information.

A socialist viewpoint expounds the principles of equality, communal sharing, and responsibility. In such a system, the focus is on the community rather than the individual which are basic tenets of individualism and capitalism. The socialist approach is, therefore, directly in contrast with such systems of thought. 

In An Inspector Calls, inspector Goole acts as the author's mouthpiece and expounds his viewpoints in his interaction with the characters. He is highly critical of those who occupy the upper echelons of the societal hierarchy, represented by the Birlings and Gerald Croft, and contrasts their lives of privilege with those who, literally and figuratively, dwell at the bottom, represented by Eva Smith and the Birling's maid, Edna.

From the outset, the inspector makes it evident that those who hold a higher status, financial or otherwise, are careless and selfish. He makes it pertinently clear that they act out of individual interest and ignore the plight of those whose destinies they control. He proves to each character whom he interrogates, that he or she has ultimately been directly responsible for Eva's suicide. 

He, for example, makes it obvious to the haughty Mr. Birling that his greed and uncaring attitude drove Eva out onto the streets.

She was out of work for the next two months. Both her parents were dead, so that she'd no home to go back to. And she hadn't been able to save much out of what Birling and company had paid her. So that after two months, with no work, no money coming in, and living in lodgings, with no relatives to help her, few friends, lonely, half-starved, she was feeling desperate. 

In similar vein, he shows how Sheila, not long after, was responsible for getting Eva to lose her job:

At last she found another job – under what name I don't know – in a big shop, and had to leave there because you were annoyed with yourself and passed the annoyance on to her. Now she had to try something else. So first she changed her name to Daisy Renton-

Gerald Croft is made to understand that he exploited Eva (now Daisy Renton) when she was most vulnerable and essentially used her. Gerald admits to having had an affair with Daisy Renton and then had to break off the relationship at a point where, as he says, he had to leave for business reasons. The poor girl was, once again, destitute. Inspector Birling informs his listeners of her sentiments after the affair:

She kept a rough sort of diary. And she said there that she had to go away and be quiet and remember ' just to make it last longer'. She felt there'd never be anything as good again for her – so she had to make it last longer.

The inspector then proves how Mrs. Birling's arrogance and conceit had negatively impacted on the poor girl's life.

She came to you for help, at a time when no woman could have needed it more. And you not only refused it yourself but saw to it that the others refused it too. She was here alone, friendless, almost penniless, desperate. She needed not only money but advice, sympathy, friendliness. You've had children. You must have known what she was feeling. And you slammed the door in her face.

Eva had fallen pregnant and sought help from a charity organisation of which Mrs. Birling is the chairperson. She advised her colleagues on the panel to turn down her request for assistance. Eva was then in a more desperate situation than ever. Mrs. Birling insistently tells the inspector that the father of Eva's unborn child should be held responsible for the poor woman's desperation.

Inspector Birling finally confronts Eric and shows how he had had a hand in Eva's final desperate act. Eric admits making her pregnant and stealing money from his father's office (fifty pounds) to support her. When she discovered that the money had been stolen, she refused it. It was then that she went to Mrs. Birling's charity organisation for help and was refused. 

In his summation, the inspector says the following in Act 3:

And be quiet for a moment and listen to me. I don't need to know any more. Neither do you. This girl killed herself – and died a horrible death. But each of you helped to kill her. Remember that. Never forget it. (He looks from one to the other of them carefully.) But then I don't think you ever will. Remember what you did, Mrs Birling. You turned her away when she most needed help. You refused her even the pitiable little bit of organized charity you had in your power to grant her. Remember what you did-

It is evident that the inspector wants everyone to acknowledge their guilt and not to forget their complicity. Their actions were what lead Eva to tragedy. The inspector asks them to remember that, probably in the hope that they will do better in the future and avoid making the same mistakes. His final words most pertinently sums up the basic principle of social responsibility:

But just remember this. One Eva Smith has gone – but there are millions and millions and millions of Eva Smiths and John Smiths still left with us, with their lives, their hopes and fears, their suffering and chance of happiness, all intertwined with our lives, and what we think and say and do. We don't live alone. We are members of one body. We are responsible for each other. And I tell you that the time will soon come when, if men will not learn that lesson, then they well be taught it in fire and bloody and anguish.

In essence, we all have a moral duty to one another, and that is to take care of each other.

Priestley uses the inspector as a vehicle for his socialist message. The inspector, by interrogating each of the family members about their involvement with Eva Smith, reveals their selfish flaws for contemplation by the audience. We see Sheila had her dismissed in a fit of pique, Mr Birling had her fired for campaigning for worker's rights. Gerald used her, Eric got her pregnant and Mrs Birling refused her charity.

The inspector's words are clear and damning. We can sense the fury of Priestley, who was a veteran of the First World War, in the final condemnation delivered to the family-

INSPECTOR: We are responsible for each other. And I tell you that the time will soon come when, if men will not learn that lesson, they will be taught it in fire and blood and anguish.

How does J. B. Priestly explore social responsibility in An Inspector Calls?

An Inspector Calls can be seen as a political allegory that presents a withering indictment of the English upper-classes between the wars. All of the wealthy, privileged characters in the play have, to some extent, failed in their duty toward the lower orders in the form of Eva Smith, and Inspector Goole has arrived on the scene to remind them of their moral, if not legal, responsibility for the young lady's tragic suicide.

Priestley's exploration of social responsibility attempts to demonstrate the extent to which the new-money elite of Edwardian society have abdicated any sense of obligation towards those less fortunate than themselves.

In bygone days, the traditional land-owning aristocracy did at least have some sense of noblesse oblige to the tenants who worked on their extensive properties. But industrial capitalists such as Arthur Birling, though more than happy to acquire the trappings of aristocratic life, are not prepared to accept the responsibilities that went with it. They see the likes of Eva Smith as expendable—as little people who can be tossed aside like a used handkerchief whenever they cease to be of use.

The social bonds that used to connect the upper and lower classes have been severed, leaving the poor and downtrodden to fend for themselves.