One of the clearest explanations of (Peircean) abduction, by David Morris.
…[A]bduction is so central it is hard to get it square in view, although Peirce does have texts that focus on it. To pierce to the quick, here are some of the things Peirce says: (1) Abduction is a procedure of rational inquiry. (2) It is a kind of inference that is insightful. (3) Abduction is neither deduction nor induction. (4) In contrast to deduction or induction, abduction adds something new to thought, namely hypotheses—and “hypothesis” is Peirce’s other name for abduction. As Peirce puts it, “the essence of an induction is that it infers from one set of facts to another set of similar facts, whereas hypothesis infers from facts of one kind to facts of another.” Peirce’s repeated example is of beans in a bag. From the fact that all the beans in the blue bag are white and that this handful of beans is from the blue bag, we can deduce that the beans in this handful are white; the deduction is certain because it adds nothing new to the facts; it just puts them a different way. If it is the case that beans taken from the blue bag keep turning up white, we conclude by induction that all beans in the blue bag are white. Here too the induction does not give us a new sort of fact, for it quantifies in a probabilistic way over facts already given about colors of beans in a bag. Abduction is different: it starts from the facts that one of the bags of beans in the room, say the blue bag, contains only white beans and that this handful of beans, which was taken from a single bag, contains all white beans; the inference by abduction is that this handful of beans is from the blue bag. Put another way, in its context, this hypothesis is the best possible explanation for the fact that the beans are white in color. Notice that the abduction yields another kind of fact: from facts about colors of beans to a hypothesis about which bag the beans are from. Sherlock Holmes uses abductive reasoning all the time, which is what astonishes Watson: it is not surprising that someone studying the color of swans might claim all swans are white, but it is surprising that someone given facts about dogs not barking in the night can confidently claim that so and so is the culprit.