Casuistry is any attempt to determine the correct response to a moral problem, often a moral dilemma, by drawing conclusions based on parallels with agreed responses to pure cases, also called paradigms.
Casuistry as a method was popular among Catholic thinkers in the early modern period, especially the Jesuits. It however was later attacked (e.g. by Pascal) as the mere use of complex reasoning to justify moral laxity; hence the everyday use of the term to mean complex reasoning to justify moral laxity.
In more recent years scholars have reconsidered casuistry; and have found many of the earlier criticisms of it to be unjustified.
Another version of the article follows--needs to be integrated (and fact-checked!).
Casuistry is an approach to applied ethics. It takes a relentlessly practical approach. Rather than applying theories, it examines cases. By drawing parallels between paradigms, so called "pure cases," and the case at hand, a casuist tries to determine the correct response (not merely an evaluation) to a particular case.
Casuistry is successful because it does not require participants in the evaluation to agree about ethical theories or evaluations before making policy. Instead, they can agree that certain paradigms should be treated in certain ways, and then agree on the similarities, the so-called warrants between a paradigm and the case at hand.
Since most people, and most cultures substantially agree about most pure ethical situations, casuistry often creates ethical arguments that can persuade people of different ethnic, religious and philosophical beliefs to treat particular cases in particular ways. For this reason, casuistry is the form of reasoning used in English Law.
Casuists have often been mistrusted as too self-serving, and their reasoning thought too inaccessible. The reasoning is often inaccessible because successful casuistry requires a large amount of knowledge about paradigms, and how parallels can be drawn from those paradigms to real life situations. In particular, the Protestant Reformation in part was a rebellion against what many saw as the self-serving causistry of the Roman Catholic Church. In modern times, there is a similar tremendous resentment against lawyers and law.
A good reference is "The Abuse of Casuistry", by Albert Jonsen and Stephen Toulmin.