(I'm not an expert, but you don't seem to have gotten any answers so far.)
Let's start from the beginning. Let $A$ be an $n\times n$ matrix of regular functions on a Zariski open subset $X\subset \mathbb{P}^1$ over $\mathbb{C}$. Then consider the system of differential equations
$$\frac{df}{dx} = A f$$
Basic theory tells us that the space of holomorphic solutions near $x_0\in X$ is $n$ dimensional. Choose a basis, and analytically continue one of these solutions along a closed path based at $x_0$. It will return to different solution. This is *monodromy*, which is measuring the "multivaluedness" of the solutions. To connect this to D-modules, let $M=O_X^n$ (sheaf of regular or holomorphic functions). This is clearly an $O_X$-module, it becomes a left $D_X$-module by letting $\partial$ act via the rule $f\mapsto f'-Af$. If you look up the literature on hypergeometric differential equations, you will see a lot of explicit examples worked out in detail.

You can jazz up the last example, by replacing $X$ by any complex manifold/smooth variety and $M$ by a vector bundle with an integrable connection $\nabla$. This gives a rule for letting vector fields act on $M$, integrability ensures that it extends to a $D$-module structure. The kernel $\nabla$ forms a locally constant sheaf, and conversely any locally constant sheaf of $\mathbb{C}$-vector spaces arises this way. So you get plenty of examples of this type on say a curve of genus at least 2. These $D$-modules are quite special in that they are both $O_X$-coherent and holonomic. For an example which is neither, take $D_X$ itself.