εν αρχη ην ο λογος
In the beginning was the word.
— Gospel According to St. John, Chapter 1, verse 1
Logic is the study of the process of reason. It is concerned with the form of argument. Take the syllogism, for a concrete and rather hackneyed example:
All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.
This can be stripped of its content and generalized to:
All x are y.
a is x.
Therefore a is y.
You can then substitute what you like for a, x and y:
All cats are carnivores.
Muddypaws is a cat.
Therefore Muddypaws is a carnivore.
Just as it is important to astronomers that their telescopes are functioning properly, it is important to people in all disciplines that their processes of reasoning are working properly. So logic is something fundamental to all science, in the broadest sense of 'science'. But nearly all logicians will tell you that what they are doing is not just philosophy of science.
Though reasoning is carried out by human minds, most logicians would say that what they are doing is not just psychology. Though it is carried out using language, and we shall be talking about grammar, syntax and semantics, most logicians would want to distinguish logic from linguistics and the philosophy of language.
From the time of Aristotle and the Prior Analytics, through to the middle ages and beyond, logic was studied in a largely verbal form, indeed λογος (logos) can mean 'word' or it can mean 'reason'. There were various forms with nice latin names like modus ponens and modus tollens, and if you want to become a logician you will need to know what those mean. But certainly from the time of Frege onwards, i.e. late 19th Century, logic has become much more symbolic, and it starts to look like mathematics, with which it has much in common. But many logicians would say that what they are doing is not just mathematics.
Or if it is mathematics, there is a big question about what mathematics is. Replies to this question are known as formalism, platonism and intuitionism. Both logic and mathematics are concerned with structure in a rather abstract way - in the case of logic it is the structure of reasoned argument, whereas with mathematics, it is often, though not always, the structure of numbers.
The stuff that looks like mathematics is called 'formal logic'. There is also a lot of stuff that isn't quite so mathematical that asks questions like: 'What are names?', 'What is truth?', and so on.
The problem with saying more about what logic is than what I have said at the beginning, is that you start to use logic to define logic. This is circular and self-referential. Logicians know that circularity and self-reference can cause problems and paradoxes. How and when they do and how and when they don't is I suppose interesting.