Vector is a mathematical object that incorporates two concepts: magnitude and direction .
Magnitude can implemented as a non-negative real number. The direction is a more complicated concept and can be implemented graphically as an arrow or numerically using a system of coordinates.
When graphical implementation is used, we have to determine a space our vector is represented (one-dimensional line, two-dimensional plane or three-dimensional space). Then the arrow from some point shows the direction of the vector. We can use the length of this arrow as an indication of the magnitude in some units. This completes the graphical representation of a vector.
Representation in the coordinate form assumes that we know the dimensionality of the vector (as above, it can be one-dimensional line, two-dimensional plane, three-dimensional space or even more abstract N-dimensional space). Then the direction and magnitude of a vector can be represented by the coordinates of its endpoint, assuming that its beginning lies at the origin of coordinates.
You can find a description of these representations, properties of and operations on vectors in many places on the Web. On my Web site Unizor Education for high school students I have a set of lectures dedicated to vectors as well. Just choose the topic "Vectors" from the front page.