What is a vector?

A quantity may be a scalar (just a magnitude, just a number) or a vector, with magnitude and direction. For example, speed is a measure of how fast something is going, but velocity is a measure of how fast something is going AND the direction it is moving in. Vectors are often represented by arrows, with the size of the arrow representing the magnitude, and the direction the arrowhead is pointing representing the vector's direction.
Vectors may be written mathematically, as the product of their magnitude and unit vector (the unit vector just contains the information about the direction a vector is pointing in).e.g.
v = v*v^{^}Vector = Magnitude * Unit vector
(The unit vector is often written like a normal vector with a "hat" pointing upwards on top of it, this is the best I could do.)