Vector
Elementary
A vector is an arrow with a length and a direction.
High school
A vector is a quantity with a magnitude and a direction. Graphically, a vector is drawn as an arrow. In coordinates, it is written as a list of numbers.
Caveat. In general vector spaces, such as spaces of functions or polynomials, there may be no meaningful notion of “magnitude” or “direction” until one is chosen.
Undergraduate
A vector is an element of a vector space: a set with vector addition and scalar multiplication satisfying the vector space axioms. Vectors are defined by what you can do with them, not by what they look like.