The term logic is used quite a lot, but not always in its technical sense. Logic, strictly speaking, is the science or study of how to evaluate arguments and reasoning. Logic is what allows us to distinguish correct reasoning from poor reasoning. Logic is important because it helps us reason correctly without correct reasoning, we dont have a viable means for knowing the truth or arriving at sound beliefs.
Logic is not a matter of opinion: when it comes to evaluating arguments, there are specific principles and criteria which should be used. If we use those principles and criteria, then we are using logic; if we arent using those principles and criteria, then we are not justified in claiming to use logic or be logical. This is important because sometimes people dont realize that what sounds reasonable isnt necessarily logical in the strict sense of the word.
Our ability to use reasoning is far from perfect, but it is also our most reliable and successful means for developing sound judgments about the world around us. Tools like habit, impulse, and tradition are also used quite often and even with some success, yet not reliably so. In general, our ability to survive depends upon our ability to know what is true, or at least what is more likely true than not true. For that, we need to use reason.
Of course, reason can be used well or it can be used poorly and that is where logic comes in. Over the centuries, philosophers have developed systematic and organized criteria for the use of reason and the evaluation of arguments. Those systems are what have become the field of logic within philosophy some of it is difficult, some of it is not, but it is all relevant for those concerned with clear, coherent, and reliable reasoning.
The Greek philosopher Aristotle is generally regarded as the father of logic. Others before him discussed the nature of arguments and how to evaluate them, but he was the one who first created systematic criteria for doing it. His conception of syllogistic logic remains a cornerstone of the study of logic even today. Others who have played important roles in the development of logic include Peter Abelard, William of Occam, Wilhelm Leibniz, Gottlob Frege, Kurt Goedel, and John Venn. Short biographies of these philosophers and mathematicians can be found on this site.
Logic sounds like an esoteric subject for academic philosophers, but the truth of the matter is that logic is applicable anywhere that reasoning and arguments are being used. Whether the actual subject matter is politics, ethics, social policies, raising children, or organizing a book collection, we use reasoning and arguments to arrive at specific conclusions. If we dont apply the criteria of logic to our arguments, we cannot trust that our reasoning is sound.
When a politician makes an argument for a particular course of action, how can that argument be properly evaluated without an understanding of the principles of logic? When a salesman makes a pitch for a product, arguing that it is superior to the competition, how can we determine whether to trust the claims if we arent familiar with what distinguishes a good argument from a poor one? There is no area of life where reasoning is completely irrelevant or wasted to give up on reasoning would mean to give up on thinking itself.
Of course, the mere fact that a person studies logic doesnt guarantee that they will reason well, just as a person who studies a medical textbook wont necessarily make a great surgeon. The correct use of logic takes practice, not simply theory. On the other hand, a person who never opens a medical textbook probably wont qualify as any sort of surgeon, much less a great one; in the same way, a person who never studies logic in any form probably wont do a very good job at reasoning as someone who does study it. This is partly because the study of logic introduces one to many common mistakes that most people make, and also because it provides a lot more opportunity for a person to practice what they learn.
It is important to keep in mind that while much of logic appears to be concerned solely with the process of reasoning and arguing, it is ultimately the product of that reasoning which is the purpose of logic. Critical analyses of the way an argument is constructed are not offered simply to help improve the thinking process in the abstract, but rather to help improve the products of that thinking process i.e., our conclusions, beliefs, and ideas.