It helps to understand irregular singularities as the merging of regular singular points, say $$(x^2-a^2)y'+y=0$$ as $a\to0$.
For nonzero $a$ the data is encoded as monodromy (constant) matrices acting on your local solutions, given by the analytic continuation along loops generating the fundamental group. 

Parts of the monodromy data makes it to the limit, namely the loops that are not cut by the merging. In the example that would be a loop encircling once both points $\pm a$, its limit encircling once the singularity $0$ and providing the monodromy part. 

Yet part of the data won't make it to the limit, and something is lost if one looks only at the monodromy of the local systems. In the example that would be the monodromy associated to any one of the loops encircling only one singularity $\pm a$.

In the scalar case, the monodromy $y\mapsto c^\pm y$ around $\pm a$ is given by $$c^\pm=\exp\frac{\pm1}{2a}$$ which gets wild as $a\to 0$. Observe though that $c^+c^-=1$ makes it to the limit as the monodromy of $x\mapsto \exp \frac{1}{x}$.

So, where has the lost data gone? And, what is the link with Stokes lines? In the above example the Stokes data is trivial, but simply considering the modified ODE $$(x^2-a^2)y'+y=x$$ gives a non-trivial example. For $a=0$ the so-called Euler's equation has a unique power-series solution $$\hat y(x) = \sum_n (n!)x^{n+1} $$ which doesn't sum as an analytic object in the usual way. By Borel-Laplace summing this series you obtain two analytic solutions, each one defined on a sector containing a half-plane, from which you deduce two sectorial local systems. The Stokes data comes from the comparison between these two local systems where the sectors overlap.The overlapping location is determined by the bissecting lines of the sectors, *i.e.* the Stokes lines.

The discussion above streeses the fact that monodromy data is not a good presentation since it doesn't pass to the limit when a regular system degenerates onto an irregular one. Moreover the distinction monodromy/Stokes data is rather artificial, since Stokes data has also a meaning as gluing of local systems. I prefer the view where everything is "Stokes data": one can always subsdivide $\mathbb P_1$ into "sectors" on which you have a trivial local system, that get compared in the pairwise intersections of said generalized sectors. In the case of a regular singularity, you can form a neighborhood around it by tiling contiguous sectors: the composition of the Stokes operators coming from crossing the corresponding overlaps attached at the singularity gives you the monondromy operator. All this data passes to the limit in cases of merging.

As the construction shows, the Stokes data is not attached to an element of the fundamental group of $X\setminus sing$, like the monodromy, but rather to the "dual" groupoid of  paths linking singular points.

A rich combinatorics comes from these considerations in the case of higher-Poincaré-rank systems (merging of $>2$ singular points). 

To read more about the above topics, look for papers by Christiane Rousseau (Montréal)

 1. (Linear systems)  Jacques Hurtubise, Caroline Lambert, and Christiane Rousseau. Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k. Mosc. Math. J., 14(2):309–338, 427, 2014.

 2. (Non-linear, but with a detailed analysis of generalizations of the examples above) a book chapter of mine https://hal-cnrs.archives-ouvertes.fr/hal-01170840
 3. (Non-linear, detailed construction and study of the generalized sectors) with Christiane Rousseau https://hal-cnrs.archives-ouvertes.fr/hal-01890315

 4. (Slightly non-linear, confluence of Stokes data in the Painlevé family) by Martin Klimes https://arxiv.org/abs/1609.05185


  [1]: https://i.sstatic.net/ZQqhI.png