Geometrical meaning of the commutator of vectors on a manifold On a manifold, vectors do not describe finite displacements, unlike in euclidean geometry, but they do describe infinitesimal displacements, so we can take two vectors, $v^a$ and $w^a$ to span an infinitesimal parallelogram.
I would like to understand how
$ v^a w^b−w^a v^b$ is what describes the "infinitesimal parallelogram" spanned by $v$ and $w$.
For example, a vector is (parallel) transported from a point $P$ to a point $P'$, along two different paths: the first $ P \rightarrow P_{1} \rightarrow P'$ consists of two infinitesimal shifts,
the second $ P \rightarrow P_{2} \rightarrow P'$ consists of the same shifts but in reverse order.
How are the movements represented ?
 A: In order to understand what (as you put it), $v^aw^b-w^av^b$ means, let me clarify some details. The commutator in differential geometry can be thought of as a measure of curvature; it doesn't tell you how the space is curved, but more or less how the two vectors you put into the commutator would be on the curved surface with respect to one another. I also want to make it completely clear that these are not just vectors, but vector fields. So, lets take two examples:
If $\mathbf{u}$ and $\mathbf{v}$ are directional derivatives, then their commutator would be, $[\mathbf{u},\mathbf{v}] = \partial_\mathbf{u}\partial_\mathbf{v}-\partial_\mathbf{v}\partial_\mathbf{u}$. But for physics (since we are on a physics forum), this is not exactly helpful...
If we wanted the commutator of two vector fields in any coordinate basis, then consider the following,
$\begin{align*}
[\mathbf{u},\mathbf{v}]f & = u^\alpha\frac{\partial}{\partial x^\alpha}\left(v^\beta \frac{\partial f}{\partial x^\beta}\right) - v^\alpha\frac{\partial}{\partial x^\alpha}\left(u^\beta \frac{\partial f}{\partial x^\beta}\right)\\
& = \left[ \left(u^\alpha v^\beta_{,\alpha} - v^\alpha u^\beta_{,\alpha}\right)\frac{\partial}{\partial x^\beta} \right]f\\
& = \left(u^\alpha v^\beta_{,\alpha} - v^\alpha u^\beta_{,\alpha}\right)\mathbf{e}_\beta,\\
\end{align*}$
where we first started out in a specific coordinate basis, but then moved to a general one by replacing the partial derivative. So, we can see that the commutator between two different vector fields really depends on out choice of coordinate frame, and is a measure of the difference between how tangent vectors behave along a curved surface.
Finally, relating back to your remark on "how is it related to an infinitesimal parallelogram," draw out a vector u that is perpendicular to the starting point of another vector v. Now, take the opposite vector of each, and start from where they ended (so, wherever u ended, place v and vice versa). If you did it correctly, then there should be a gap between the two ending vectors. This gap is literally $[\mathbf{u},\mathbf{v}]$. That is why it can be though of as an "infinitesimal parallelogram."
