Someone unperturbed stays calm when something has happened that would upset, worry, or annoy most people. The important part is the comparison hiding inside the word: you only call a person unperturbed when there was a reason to be bothered and they were not. Nobody is 'unperturbed' while eating breakfast quietly at home — but a speaker who keeps talking smoothly while the microphone dies, or a cat that sleeps through fireworks, is unperturbed. It is a fairly formal, written word; you find it in news writing and stories more than in conversation, where people say 'he didn't seem bothered' or 'she stayed cool'. Two common shapes: 'unperturbed by' something ('unperturbed by the criticism'), and standing alone after a verb like seem or appear, or as a tag on a sentence ('Unperturbed, she carried on'). There is often a small flavour of admiration or of dry humour in it — the writer is quietly pointing at the gap between how bad things were and how untroubled the person looked. Depending on the story, that can read as impressively steady, or as slightly comic, or even as a hint that the person did not care as much as they should have.