A vector is a property of an object or action that acts on an object which has a magnitude and a direction. The easiest vector to understand is the velocity vector, which is a property. The speed of an object is the magnitude of the velocity vector. The direction of travel is the direction of the vector.
Graphically, we represent the vector by the magnitude next to the unit vector, an arrow, which signifies the direction. Algebraically we express the vector as the magnitude multiplied by the unit vector, where we represent the unit vector by a symbol with a "hat". The vector itself we represent by printing in bold.
Alternatively, when handwritten, the vector can be represented by a symbol with an arrow above it.
An example of a vector that acts on an object is a force vector. The magnitude of the force is the strength.