Teaching Excellence for a Fee
In fifth-century Athens, democratic politics made public speaking the route to influence. The Sophists were itinerant professional teachers who met the demand, offering instruction in rhetoric, argument and practical excellence, and charging for it.
Two things about them scandalised. Charging fees was thought to demean wisdom, and their subject was not nature but human affairs: law, convention, persuasion, politics. Their central distinction is nomos and physis: convention against nature. Which of our arrangements are natural and inevitable, and which are merely human agreements that could have been otherwise? Once asked, the question does not go away, and it is the origin of social and political philosophy.
Protagoras gave the famous formulation: "Man is the measure of all things, of things that are, that they are, and of things that are not, that they are not." Read strongly, this is relativism: truth is relative to the perceiver, and if the wind seems cold to me and warm to you, it is cold for me and warm for you. He also professed agnosticism about the gods, and claimed to teach both sides of any question, which is either intellectual honesty or the abandonment of truth, depending on your view.
Gorgias pushed further in a work arguing three theses in sequence: nothing exists; if anything existed it could not be known; if it could be known it could not be communicated. Whether this is serious philosophy or a demonstration of rhetorical power is still debated, and that ambiguity is itself characteristic.
The reputation is partly deserved and partly Plato's doing. Some Sophists genuinely taught that the stronger argument could be made to appear the weaker regardless of truth, which is what "sophistry" now means. But Plato had a specific reason to attack them: Socrates looked exactly like a Sophist — arguing in public, questioning conventions, gathering young followers — and was executed partly on that account. Distinguishing his teacher from them was an urgent task, not a neutral one.
The permanent contributions are real: the turn to human affairs, the nomos/physis distinction, systematic attention to language and argument, and the demand that convention justify itself.
Common pitfall: taking Plato's portrait at face value. He is a hostile witness with a strong motive, and the Sophists' own writings largely do not survive. What we call sophistry may be Plato's characterisation more than their practice.