The term vector can have two different meanings.
A. Most people particularly physics students & learn vectors as direction and magnitude. This meaning has an alternate representation/ viewpoint as coordinates in space (2 dimensions, 3 dimensions or higher).
In the introductory linear algebra courses I am familiar with, most examples of vectors are of this first type, so fit well as a column inside a matrix.
B. From a linear algebra & higher math perspective, the term vector has a much broader, more abstract meaning. First define a vector space, then as you mentioned a vector is an element in that space.
For example, you can have a vector space of functions,
such as the vector space consisting of all polynomials of degree $3$
or less (including the zero polynomial). Addition and scalar multiplication
are defined in the standard way.
Another example is the space of all continuous
real valued functions defined on an interval $[a,b]$.
(We can define an inner product on this space by
$\langle f, g \rangle = \int_a^b f g \, dx$.)
Some clarification as to where you're coming from would be helpful to better tailor the answer to your subject. Physics, Calculus/Math of 2 or 3 dimensions, linear algebra, etc.