Definition. We say that $T\in \mathcal{D}'(\Omega)$ has $\textbf{order}$ $m$ if $m$ is the minimum integer for which there exists a constant $C>0$ such that $$|\langle T,f\rangle_{\mathcal{D}'\leftrightarrow\mathcal{D}}|\leq C\left(\sum_{|\alpha|\leq m}\|D^\alpha f\|_{\sup(\Omega)}\right),\quad\forall f\in\mathcal{D}(\Omega)$$ If a distribution does not have order $m$ for any $m\in\mathbb{N}$, then it has infinite order.
I understand the definition, however, not sure what this means in more intuitive sense. I see the right-hand side is somewhat semi-norm on $\mathcal{D}(\Omega)$, so when the order of the distribution is higher, then it seems like the distribution is "wilder" in some sense. I will appreciate it if someone can help me to understand the motivation behind of defining the order of distribution.
Intuitive, but imprecise:
The order of a distribution $u$ is the highest order of derivative of the test function $\varphi$ that the value $\langle u, \varphi \rangle$ depends on.
An easy case: the order of $\partial^\alpha \delta$ is $|\alpha|$.