Logic is
reason; the original
Greek term means 'pertaining to
logos', where 'logos' apparently means something like 'reason', 'proportion', 'order'; it is also the Greek word for 'word' - for instance, in the Biblical phrase '
In the beginning was the word'.
Logic is concerned with the truth of statements about the world.
It is based on the insight that we can derive nontrivial information on the world by means of thought: that we can become aware of truths about the world purely by combining information in our minds. Such combination is called reason and an instance of it is called a logical argument.
We all use logic in daily life, we all think and argue. In ancient Greek and Roman times, reason was considered an advanced skill, essential to a good man's education; the extreme emphasis placed on deductive argument by thinkers such as Socrates and the sophists, and the ground-breaking mathematical work by Euclid and others, have been major forces in shaping our Western civilization.
To the Greeks and Romans, there wasn't a clear-cut distinction between rhetorics (the art of speech/debate) and science (the discovery of truths about the world). Perhaps this is really the distinction between deduction and induction.
The proper analysis of logical arguments is essential to rhetorics, and our knowledge on this matter really hasn't advanced all that much in the last 2000 years; e.g. we still use Roman terminology to describe common logical fallacies. But in science, formal reasoning has seen immense progress: mathematical reasoning has advanced far beyond the simple geometry and calculus known to the Greeks. Mathematics is also applied to logic itself: probability theory is a way to model reasoning with quantifiable uncertainty, formal mathematical logic approaches qualitative reasoning as a computational process.