what are conormal distributions? According to the first answer in this post, a conormal distribution $u$ on a manifold $X$ relative to a (closed, embedded) submanifold $Y$ is an element of a Banach (or Hilbert) space $H$ such that for any positive integer $k$, we have
$$
V_1\cdots V_k u \in H
$$
where $V_i$ denotes a vector field that is tangent to $Y$ (smooth and unconstrained away from $Y$).
I would like to better understand these objects. 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" ? 
 A: 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. 
A: At first, in the definition, we are taking derivatives along directions tangent to this submanifold when we are standing on it, and we could taking derivatives along any directions elsewhere.
Thus, an object is conormal to a surface (submanifold) means that, along this submanifold, this distribution is smooth, and possible non-smoothness arise in the normal directions. 
And the regularity it remains after derivative could depend on the context, as the answer of Rafe Mazzeo indicated. He took some weighted $L^\infty$, and in Volume 3 of H\"ormander, page 100, definition 18.2.6, he took some more involved Besov space as the regularity we want to keep.
