The Inspector is a ghost, or a personification of the Birlings' (and Gerald's) consciences.
Think about it — The Reveal raises the point that no-one has actually seen this "Goole" person, which raises the suspicion as to if the Inspector is actually real or not. Also, Sheila mentions that the Inspector seemed to know everything before the Birlings even revealed their secrets. How could he have possibly known about Gerald's affair and Eric's drunken night out during which he impregnated 'Eva'? Simple: If he was part of their consciences, then it would make perfect sense for him to know.Also, notice how Sheila and Eric are affected by it and start to change their views? That wasn't change, it was part of their conscience all along, but it took a direct confrontation with their consciences in the form of Inspector Goole to realise it. Mr Birling, Mrs Birling and Gerald, on the other hand genuinely believe that they had nothing to do with it and weren't responsible, and therefore were not affected.
As his name suggests, Inspector Goole is a literal, Arabian-style ghoul.
Eva Smith was going to be his next meal after he she committed suicide, but a policeman heard the noise and rushed to investigate. Inspector Ghoul (as we`re going to call him for now) killed the inspector and ate him. He then read Eva's diary, and in a flood of pity for her went off to torment the family who led to her suicide in the form of the murdered inspector. The policeman who arrived later was the actual investigator who found Eva's corpse and diary. They're all dead.
The Birlings actually all died of food poisoning at dinner that night, and this is their personalized Hell. In a similar vein, the Birlings are in Purgatory.
They're doomed to repeat the evening over and over until all of them can accept responsibility for causing Eva's death, hence why the events seem to be beginning again when the play ends. Inspector Goole dunnit.
He's the Villain of Another Story - he poisoned Eva Smith, having first dug up dirt on the Birlings and forged a diary to make it look like suicide. He makes sure they all have guilty consciences, tips off the police, and walks away having got away with the perfect crime. There were actually four (or maybe five) different girls.
- As Gerald points out, Goole very carefully only showed the photo to one of them at a time, and they only have his word for it that these were all the same girl. Eva Smith, the girl at the clothing store, Daisy Remington, Sarah, and the girl Mrs Birling turned down could have all been five different girls (though the fact that the pregnant girl gave her name as "Mrs. Birling" does imply that she was Sarah, so more likely there were four girls). Only one of them (most likely Sarah/the pregnant girl) committed suicide that night, but Goole decided to take the opportunity to call all the Birlings to account for the various lives they'd harmed.
Inspector Goole is Eva Smith.
- Eva took on another identity, Inspector Goole, when she realised all the people that screwed her over were related to each other. She used her knowledge of their misdeeds to bait them into confessing their misdeeds, and then killed herself once she left the Birling residence.
- As for why they didn't recognise her? She cut her hair and deliberately deepened her voice.
Benoit Blanc is a similar entity to Inspector Goole, only Played for Laughs.
Think about it: they're both detectives, though not associated with any police service. Both have conspicuous accents for the time and place they're in, accents that were/are associated stereotypically with lower-class, less intelligent people. They each help underprivileged women to extract justice from their wealthy tormentors. Maybe they're both Avenging Angels of the same breed.