Vectors are typically described as a quantity that has both a magnitude (length) and direction. Exactly what does this mean? Some simple examples illustrate the concept best.
2 miles north
12 inches to the right
1 kilometer northeast
Why are vectors useful? Because they make all sorts of movement and spacial behavior a lot easier to understand and implement in code. Vectors can be added, scaled, rotated, and pointed at things. They also form the foundation of more sophisticated programming subjects like physics simulations. Most importantly, when you get the hang of them, they are huge amounts of fun.