Vectors are useful for tracking the movement of an object and/or its velocity. This is a very important topic in a world where we have robot cars, auto-pilot functions in aeroplanes and other objects moving in an automated way.
Vectors are fundamentally about representing how far up/down and sideways you go. Vectors don't care what order you make the movements, only where you end up.
What is a Vector? It Represents the Total Change. (YouTube)
Abstract Vector Problems
Once you are familiar with the basics it is time to start trying more abstract questions.
Vectors and Lines
Just like in regular algebra a line is useful to represent an infinity of possible solutions. More mathematics skills are needed to cope with this.
Vector Equation of a Line (YouTube)
Vectors and Operations
As we get more advanced we can start doing "operations" with vectors that are broadly similar to things like multiplying numbers.
Dot Product Written Different Ways (UP4) (YouTube)
Here is a link back to the UPmaths homepage.