AKA Theoretical Induction, Argument to the best explanation, and Hypothetico-deductive method.
In the fields of philosophy and logic, abduction is the inference of an explanation from a set of observations.
When using inductive logic you can predict what might happen next based on what has happened before. ("The sun came up yesterday, and the day before, and the day before -- I bet it will rise tomorrow too.") But you can't explain why it happened. For example, if I find that when flipping a coin heads comes up 60% of the time I can say, using induction, that in the next hundred throws heads will probably come up about 60 times because one side of the coin is slightly heavier than the other. This is a hypothesis; the creation of an explanation for induced data.
So, how do you justify these sorts of explanations? So far we have found only one way--called abduction. Under abduction, the way to justify the hypothesis is that it is the best explanation we have to explain the data. This could be considered a problem, because we don't seem to have a logical method of scientific advancement. Any hypothesis is generated by creativity, not logic. They are justified by logic, but a theory doesn't come about because the laws of formal logic dictate it -- it comes about because we came up with (e.g., imagined) a lot of possible stories to explain it, and then tested them.
The rules of logic and math do not require us to believe that the earth revolves around the sun. We simply came up with multiple stories to explain the solar system, and it turns out that the one that works the best is the one with the sun at the center. There could be a set of empirical data that would disprove this. And given all the data we had when we made this theory, we could still have missed the fact that the Earth revolves around the Sun if some bright person hadn't figured out that that was a possibility. It was necessary for someone to have a creative, inventive thought for us to figure this out. No amount of deductive logic alone could have brought us to this conclusion.
Scientific theories, then, are based not on deduction but on induction -- opening the way for Skepticism, since you can only be 'probably right'.