What is a vector? Well, what most physics students in high-school learn about is that they are mathematical objects that have both “magnitude and direction”, i.e. they’re essentially arrows in space.
Note: they can have an arbitrary number of “components” to them, so there really is such a thing as a “56 dimensional vector” existing in 56 dimensional space! (though, when will that ever be useful?)
Vectors are used to represent a lot of things in the physical world, so the results and Theorems concerning them really are important, as they might give you some more insight into what’s going on in the real world.
Some things vectors are used to represent: Forces, velocity and acceleration, Torque, angular velocity, Current, and more.