Conjugacy classes -- how to generate them for a list to be sorted? In another thread I had brought up the notion of sorting a list of four randomly scrambled items.
It was mentioned that they can be broken down into 5 conjugacy classes:
(), (12), (123), (12)(34) and (1234)
Can anyone explain how these work or if there is a general way to list all possible conjugacies? For instance, what about a list of size 6?

I write this answer only to make sure that OP realises the connection between the problem stated and the formulation. I think this however could be closed as exact duplicate.

The notion of sorting $n$ items  you're talking about is formally called the permutations of $n$ symbols. The notion of conjugacy discussed in the answer corresponds to the action of group on itself by conjugation. 

Well, you are asking for the number of conjugacy classes in a symmetric group of order $n$. Yes, there is a nice description. 
I'll recall the main result while I'll let you go through the details in an exactly same answer $^\dagger$ I had written over here.
Main result:
The number of conjugacy classes in $S_n$ equals the number of partitions of $n$.

We'll give a way to list an exhaustive set or representatives for the conjugacy classes.
Write down all the additive partitions of $n$. To each partition, associate a representative as follows. 
For each number appearing in the partition, attach with it a disjoint cycle of that length. The product of all such cycles represents a unique conjugacy class. It is best illustrated by an example for $4$:
$$\begin{align*}Id &\cong 1+1+1+1\\(1234)&\cong 4\\(12)(34) &\cong 2+2\\ (34) &\cong 1+1+2(\text{since (1) and (2) are omitted in this notation})\\(123)&\cong 3+1\end{align*}$$ 

$\dagger$ This answer of mine deals with exactly this question.
