Here is a very incomplete description of conormal distribution. Let $\dim(M)=l+n$, and $\dim(X)=l$. Locally use a partition of unity we may assume $M=\mathbb{R}^{l+n}$ and $X=\mathbb{R}^{n}$. Then a conormal distribution $u$ is of the form $ u=\int e^{i\xi\cdot z}a(x,\xi)d\xi $ where $a(x,\xi)d\xi$ is viewed as a density and usually called as the `left symbol' of the conormal distribution. As a distribution, $u$ acts on function $\phi(x,z)$ by integration: $ \langle u,\phi\rangle=\int e^{i\xi\cdot z}a(x,\xi)\phi(x,z)d\xi dx dz $ And you may think of this in terms of (inverse) Fourier transform near the diagonal if you wish.
For your questions, you asked:
Hence I was wondering whether somebody could suggest a good example for a conormal distribution, or more detailed explanation of what these distributions "look like" ?
My level is too low to give a serious answer for either of the questions. I believe you need to read Hormander's book (Vol III) or Melrose's notes (Microlocal analysis) to have a better understanding on this. There is another set of notes by Santigo Simanca, who is a student of Melrose, but it is not publicly available online.