$\newcommand{\dd}{\partial}$In the sense that you're asking, velocity of a point particle is a vector, and any object that "measures a velocity and returns a scalar" is a covector.
Though there's a meaningful mathematical distinction between a vector space $V$ and its dual space $V^{*}$, it's probably easier to see the distinction between a vector-valued function and a covector-valued function, because (as lists of component functions) they transform differently under change of coordinates.
For simplicity, assume we're in a plane region, equipped with coordinates $(x_{1}, x_{2})$. Rather than writing vectors as column matrices and covectors as row matrices, introduce the coordinate vector fields
$$
\frac{\dd}{\dd x_{1}},\qquad
\frac{\dd}{\dd x_{2}},
\tag{1a}
$$
and the coordinate one-forms
$$
dx_{1},\qquad
dx_{2}.
\tag{1b}
$$
A smooth vector field is a "linear combination" of the coordinate fields with smooth functions coefficients, e.g.
$$
X = X^{1}\, \frac{\dd}{\dd x_{1}} + X^{2}\, \frac{\dd}{\dd x_{2}}.
\tag{1c}
$$
A smooth covector field (or smooth one-form) is a "linear combination" of the coordinate one-forms with smooth functions coefficients, e.g.
$$
\xi = \xi_{1}\, dx_{1} + \xi_{2}\, dx_{2}.
\tag{1d}
$$
If $(y_{1}, y_{2})$ is another coordinate system, the chain rule gives
\begin{align*}
\frac{\dd}{\dd x_{1}}
&= \frac{\dd y_{1}}{\dd x_{1}}\, \frac{\dd}{\dd y_{1}}
+ \frac{\dd y_{2}}{\dd x_{1}}\, \frac{\dd}{\dd y_{2}}, &
\frac{\dd}{\dd x_{2}}
&= \frac{\dd y_{1}}{\dd x_{2}}\, \frac{\dd}{\dd y_{1}}
+ \frac{\dd y_{2}}{\dd x_{2}}\, \frac{\dd}{\dd y_{2}};
\tag{2a} \\
dx_{1}
&= \frac{\dd x_{1}}{\dd y_{1}}\, dy_{1}
+ \frac{\dd x_{1}}{\dd y_{2}}\, dy_{2}, &
dx_{2}
&= \frac{\dd x_{2}}{\dd y_{1}}\, dy_{1}
+ \frac{\dd x_{2}}{\dd y_{2}}\, dy_{2}.
\tag{2b}
\end{align*}
Writing
$$
X = Y^{1}\, \frac{\dd}{\dd y_{1}} + Y^{2}\, \frac{\dd}{\dd y_{2}},
$$
substituting (2a) into (1c), and equating components gives the transformation rule for vector fields:
$$
Y^{1}
= X^{1}\, \frac{\dd y_{1}}{\dd x_{1}}
+ X^{2}\, \frac{\dd y_{1}}{\dd x_{2}},\quad
Y^{2}
= X^{1}\, \frac{\dd y_{2}}{\dd x_{1}}
+ X^{2}\, \frac{\dd y_{2}}{\dd x_{2}}.
\tag{3a}
$$
Similarly, writing
$$
\xi = \eta_{1}\, dy_{1} + \eta_{2}\, dy_{2},
$$
substituting (2b) into (1d), and equating components gives the transformation rule for one-forms:
$$
\eta_{1}
= \xi_{1}\, \frac{\dd x_{1}}{\dd y_{1}}
+ \xi_{2}\, \frac{\dd x_{2}}{\dd y_{1}},\qquad
\eta_{2}
= \xi_{1}\, \frac{\dd x_{1}}{\dd y_{2}}
+ \xi_{2}\, \frac{\dd x_{2}}{\dd y_{2}}.
\tag{3b}
$$
Classically, a vector field is a collection of functions associated to a coordinate system that transform like (3a), and a one-form is a collection of functions that transforms like (3b). And that, in a sense, is the distinction between vectors and covectors.
Admittedly, the coordinate vector field and coordinate one-form notation is merely a formalism, just like column and row matrices. The tie-in with the chain rule, however, may make this formalism appealing, even compelling.
The connection with matrix notation comes from expressing (3a) and (3b) in terms of matrix products. For brevity, write
$$
a_{j}^{i} = \frac{\dd y_{i}}{\dd x_{j}},\qquad
b_{j}^{i} = \frac{\dd x_{i}}{\dd y_{j}},
$$
so that the matrices
$$
A = \left[\begin{array}{@{}cc@{}}
a_{1}^{1} & a_{2}^{1} \\
a_{1}^{2} & a_{2}^{2} \\
\end{array}\right] = \frac{\dd(y_{1}, y_{2})}{\dd(x_{1}, x_{2})}
$$
and
$$
B = \left[\begin{array}{@{}cc@{}}
b_{1}^{1} & b_{2}^{1} \\
b_{1}^{2} & b_{2}^{2} \\
\end{array}\right] = \frac{\dd(x_{1}, x_{2})}{\dd(y_{1}, y_{2})},
$$
the Jacobian of the change of coordinates from $x$ to $y$ and the Jacobian from $y$ to $x$, are inverses.
If we identify
$$
\frac{\dd}{\dd x_{1}} = \left[\begin{array}{@{}c@{}}
1 \\
0 \\
\end{array}\right],\qquad
\frac{\dd}{\dd x_{2}} = \left[\begin{array}{@{}c@{}}
0 \\
1 \\
\end{array}\right],\qquad
X = \left[\begin{array}{@{}c@{}}
X^{1} \\
X^{2} \\
\end{array}\right],
$$
and
$$
dx_{1} = \left[\begin{array}{@{}cc@{}}
1 & 0 \\
\end{array}\right],\qquad
dx_{2} = \left[\begin{array}{@{}cc@{}}
0 & 1 \\
\end{array}\right],\qquad
\xi = \left[\begin{array}{@{}cc@{}}
\xi_{1} & \xi_{2} \\
\end{array}\right],
$$
then (3a) and (3b) read, respectively,
$$
\left[\begin{array}{@{}c@{}}
Y^{1} \\
Y^{2} \\
\end{array}\right]
= A \left[\begin{array}{@{}c@{}}
X^{1} \\
X^{2} \\
\end{array}\right]
$$
and
$$
\left[\begin{array}{@{}cc@{}}
\eta_{1} & \eta_{2} \\
\end{array}\right]
= \left[\begin{array}{@{}cc@{}}
\xi_{1} & \xi_{2} \\
\end{array}\right] B.
$$
Particularly,
$$
\xi(X) = \xi_{1} X^{1} + \xi_{2} X^{2} = \eta_{1} Y^{1} + \eta_{2} Y^{2};
$$
the pairing between a one-form and a vector field is independent of the coordinate system, and consequently represents a "physical" or "geometric" concept.
