Sophism
Sophism was originally a term for the techniques taught by a highly respected group of philosophy and rhetoric teachers in ancient Greece. The common modern usage is not necessarily representative of the beliefs of the original Sophists, except that they generally taught Rhetoric. The Sophists are known today only through the writings of their opponents (specifically Socrates, Plato, and Aristotle), which makes it difficult to formulate a complete view of the Sophists' beliefs.
Origins
The meaning of the word sophist (gr. sophistès) has changed greatly over time. Initially, a sophist was someone who gave sophia to his disciples, i. e. wisdom made from knowledge. It was a highly complimentary term, applied to early philosophers such as the Seven Wise Men of Greece.
Eventually, "sophist" came to refer to a school of philosophy whose practitioners taught the arts of debate and rhetoric. Due to the importance of these skills in the litigious social life of Athens, acclaimed teachers of such skills often commanded very high fees. The practice of taking fees, coupled with the willingness of many practitioners to use their rhetorical skills to pursue unjust lawsuits, eventually led to a decline in respect for this school of thought.
Protagoras is generally regarded as the first sophist. Other leading 5th-century sophists included Gorgias and Prodicus. Socrates was perhaps the first philosopher to significantly challenge the Sophists.
By the time of Plato and Aristotle, "sophist" had taken on negative connotations, usually referring to someone who used rhetorical sleight-of-hand and ambiguities of language in order to deceive, or to support fallacious reasoning. Socrates, Plato, and Aristotle all challenged the philosophical foundations of sophism. Eventually, the school was accused of immorality by the state.
In modern philosophical usage, sophistry is a derogatory term for rhetoric that is designed to appeal to the listener on grounds other than the strict logical validity of the statements being made.
The Sophists held a relativistic view on cognition and knowledge. Their philosophy contains criticism of Religion, law and ethics. Though many sophists were as religious as their contemporaries, some held atheist or agnostic views.
Unfortunately most of the original texts written by the sophists have been lost, and modern understanding of sophistic movement comes from analysis of Plato's writings. It is necessary to keep in mind that Plato and the sophists had big ideological differences, and Plato might have benefited from modifying or slanting the original sophistic arguments when he presented them in his writings (ironically, a sophistic technique at work), or may even not have fully understood their arguments himself. An excellent book on the topic is "The Sophistic Movement" by G. B. Kerferd.
In the Roman Empire, sophists were just professors of rhetoric. For instance, Libanius, Himerius, Aelius Aristides and Fronto were considered sophists in this sense.
Reconstruction of Sophist philosophy
In traditional logical argument, a set of premises are connected together according to the rules of logic, and lead therefore to some conclusion. When someone criticizes the argument, they do so by pointing out either falsehoods among the premises or logical fallacies, flaws in the logical scaffolding. These criticisms may be subject to counter-criticisms, which in turn may be subject to counter-counter-criticisms, etc. Generally, some judge or audience eventually either concurs with or rejects the position of one side, and thus a consensus opinion of the truth is arrived at.
The essential insight of sophistry is that the actual logical validity of an argument is irrelevant; it is only the ruling of the audience which ultimately determine whether a conclusion is considered "true" or not. By appealing to the prejudices and emotions of the judges, one can garner favorable treatment for one's side of the argument, and cause a factually false position to be ruled true.
The philosophical Sophist goes one step beyond that, and points out that since it was traditionally accepted that the position ruled valid by the judges was literally true, any position ruled true by the judges must be considered literally true, even if it was arrived at by naked pandering to the judges' prejudices — or even by bribery.
Various (perhaps even most) politicians employ sophistry, as well.