In one sense, you can say that a vector is simply an object with certain
properties, and it is neither a row of numbers nor a column of numbers.
But in practice, we often want to use a list of $n$ numeric coordinates to describe
an $n$-dimensional vector, and we call this list of coordinates a vector.
The general convention seems to be that the coordinates are listed in the
format known as a *column* vector, which is (or at least, which acts like)
an $n \times 1$ matrix.

This has the nice property that if $v$ is a vector and $M$ is a matrix
representing a linear transformation, the product $Mx$, computed by the usual
rules of matrix multiplication, is another vector (specifically, a column vector)
representing the image of $v$ under that transformation.

But because we write mostly in a horizontal direction and it is not always
convenient to list the coordinates of a vector from left to right.
If you're careful, you might write

$$ \langle x_1, x_2, \ldots, x_n \rangle^T $$

meaning the *transpose* of the row vector $\langle x_1, x_2, \ldots, x_n \rangle$;
that is, we want the convenience of left-to-right notation but we
make it clear that we actually mean a column vector
(which is what you get when you transpose a row vector).
If we're *not* being careful, however, we might just write
$\langle x_1, x_2, \ldots, x_n \rangle$
as our "vector" and assume everyone will understand what we mean.

Occasionally we actually need the coordinates of a vector in row-vector format,
in which case we can represent that by transposing a column vector.
For example, if $u$ and $v$ are vectors (that is, column vectors), then the
usual inner product of $u$ and $v$ can be written $u^T v$, evaluated as
the product of a $1\times n$ matrix with an $n \times 1$ matrix.
Note that if $u$ is a (column) vector, then $u^T$
really is a row vector and can (and should) legitimately be written as
$\langle u_1, u_2, \ldots, u_n \rangle$.

This all works out quite neatly and conveniently when people are careful
and precise in how they write things.
At a deeper and more abstract level you can formalize these ideas as shown in
another answer.
(My answer here is relatively informal, intended merely to give a sense of why
people think of the column vector as "the" representation of an abstract vector.)

When people are *not* careful and precise it may help to say to yourself sometimes
that the transpose of a certain vector representation is *intended* in a
certain context even though the person writing that representation
neglected to indicate it.