Is there a good overview of conjugacy classes?

During the lecture of Algebra we had the topic „Conjugacy classes of $A_n$ and $S_n$“ and I somehow don‘t see through it. So I know that if we have a group acting on itself, then two elements are in the same conjgacy class if they are conjugate. Then we also had a statement saying that two elements are conjugate in $S_n$ if they have the same partition of $n$. But I really don‘t see through it, so I don‘t see where this can be used and how to work with. Does maybe someone know if there are some useful overview about this topic or are there some nice examples?

This would be very helpful.

Thank you for your help

Let's simplify the case first. Suppose we have a $i$-cycle $(a_1\; \cdots\;a_i)$ of $S_n$ and an arbitrary permutation $\sigma \in S_n$. Then conjugating $(a_1 \; \cdots \; a_i)$ by $\sigma$ gives $$\sigma \circ (a_1 \; \cdots \; a_i) \circ \sigma^{-1} = (\sigma(a_1) \; \cdots \; \sigma(a_i))$$ (I will leave the verification to you; it is ruotine.) For the general case, we simply recall that every permutation can be expressed as a product of disjoint cycles. Hence, we can see quickly that conjugating a permutation preserves its cycle type (which corresponds naturally to a partition of $n$)