A mathematical concept that can be used to express position, direction or velocity and which can simplify or outright trivialise spatial problems.
Why do we care about linear algebra?
Linear algebra is the study of vectors. If your game involves the position of an on-screen button, the direction of a camera, or the velocity of a race car, you will have to use vectors. The better you understand linear algebra, the more control you will have over the behavior of these vectors.
— from the tutorial you're about to see the link to
In several programming languages, "Vector" is also used to refer to a one-dimensional dynamically sized array, rather than the mathematical Euclidean vector explained above. Any questions discussing the array version most likely belong in Stack Overflow or otherwise should not be given this tag.