This is in the case of a conductor:
I want to know what happens in the case of insulators.
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I know there is at least one almost exact duplicate to this question (mentioned in the comment below the post) but I was half way through writing this answer when I came across it.
The best way to explain what happens in the case of an insulator is to understand that there are no free charges in an insulator (or dielectric).
The Dielectric only contains bound charges in atoms (net charge=0). However the dielectric can have an electric field and this is the case when a dielectric is placed between the plates of a capacitor (which produces a uniform electric field between the capacitor plates).
Placing the dielectric between the plates of the capacitor will cause polarization and the electron clouds surrounding the nuclei will be displaced a tiny amount due to the electric field of the capacitor (or anything else that produces an electric field):
As you can see from the image below the atoms within the dielectric behave like tiny electric dipoles where the electron cloud I mentioned before is represented by the negative sign and the positive nucleus as the positive sign. These dipoles align in an electric field since an electric field exerts a force on electric charge:
For comparison; the image below shows how the electric field is affected by both an insulator and a conductor:
I will just emphazize on the comment I made below your post that it is important that you understand why the conductor distorts the electric field in that way.