Vectors are the basis of many geometry, mathematical, and design systems. Simply put, a vector consists of two things: a direction, and a value. The direction is specified as relative to some frame of reference. You can think of a compass referencing the poles, or the XY axis in a graph.
The value can be used to represent a lot of things. The most easy to understand is length. You can picture a vector as an arrow. The arrow has a certain length (the value) and a direction. As soon as you try to define the direction of the arrow, you will see that you need some kind of reference.
Ok, enough about the basics of vectors. We are talking about design here.
You might have heard of a vector drawing program?
Well, that is what most designers use. Back in the days of windows paint as the primary way to make computer drawings, all we could do is manipulate pixels. That has some huge drawbacks like scaling and making changes.
Vectors are a layer of programming that is run on top of the actual pixel manipulation. By using vectors, we can scale images without any loss of quality.