I am trying to understand the ideal class group. I have seen a few different derivation of this.
Sometimes i read about the ideal class group of a ring of algebraic integers ($\mathcal{O}_K$) and sometimes i read about the ideal class group of an imaginary field $ K =\mathbb{Q}(\sqrt{m})$, with $m<0$. Is there a difference? Because i thought, that a field only has two ideals and hence the definition of ideal class group over a a field does not make much sense to me.
You are correct that a field has only two ideals. When we speak of the ideal class group of an algebraic number field, we are speaking of the ideal class group of the ring of integers in that field.