Start with charges against Socrates, of being a cosmologist and a sophist, as he himself points out in the Apology: "There is a clever man called Socrates who has theories about the heavens and has investigated everything below the earth, and can make the weaker argument defeat the stronger." (Grippe, p11-29) "Socrates is committing an injustice, in that he inquires into things below the earth and in the sky, and makes the weaker argument defeat the stronger, and teaches others to follow his example." (Section A, 19b) "if you have heard anyone say that I try to educate people and charge a fee, there is no truth in that either…" (Grippe, p11-29)
You already know that cosmologists, or natural philosophers, were men who sought natural explanations for material processes, as opposed to supernatural explanations. They rejected Poseidon's trident blow as an explanation for earthquakes, much like we eventually come to favor the natural explanation for thunder over "God is bowling" or "God is wrestling with the devil." Those are both examples of a culture drawing on the divine for answers to questions about natural processes. But cosmologists rejected the old gods.
Sophists were philosophers who turned their attention away from investigation of the natural world and adopted a much more practical approach to the use of philosophy. Sophists traveled wherever they could find a client, and they earned their living by teaching such arts as rhetoric and argumentation, skills that people eagerly paid to learn and use (and often abuse). (Grippe, p11-29) Protagoras (481-411 BC), earliest of the Sophists, is credited with saying "man is the measure of all things." Protagoras will be expelled from Athens in 415 BC. Sophists pointed out that while physis (nature) is controlled by an unchanging set of laws that remain the same wherever you go, nomos (custom, or man-made law) is arbitrary and changes from state to state. This empowered men to think that they could shape the world and be masters of their own fates. Originally, the goal of sophistry was defined as "teaching excellence or virtue (arete)" primarily through public speaking (the way any Greek of the 5th century made his way). (Grippe, p11-29)
But sophistry became known as unscrupulous teaching that leads to unscrupulous behavior. Because Athenian upperclass men made their way in society by learning the skills of rhetoric and argumentation - remember you were your own lawyer and press agent - then the sophists had more students than they could handle. Everyone wanted to learn this new-fangled way of manipulating facts so they could come out ahead. After all, isn't that what rhetoric and argumentation is all about? Manipulating your audience? Knowing the needs and likes of your audience so that you satisfy them? So sophists sold their wares to the highest bidder, and taught whatever skills would lead to success. So sophists eventually came to be known for teaching young men to successfully argue anything two ways, and being able to make any argument, right or wrong, win. (Goldman, 33p)