How are the equivalence classes made of orbits defined I am reading through Freileigh, and I'm curious to know if I am understanding the definition correctly. 
Each Permutation of a set $A$ determines a natural partition of $A$ into cells with the property that $a, b\in A$ are in the same cell if and only if $f^n(a)=b$ for some $n$ that's an element of $\mathbb Z$. 
Is this saying for every permutation of the set $A$, where each permutation is being represented by $f^n$ for integers $n$? 
Say I had two permutation of $S_4$: 
$f^0$ and $f^1$
$f^0 = (1\mapsto1,\ 2\mapsto2,\ 3\mapsto3,\ 4\mapsto4)$ 
$f^1 = ( 1\mapsto2,\ 2\mapsto4, \ 3\mapsto1,\ 4\mapsto3 )$
The orbits of $f^0 = \{1\} \{2\} \{3\} \{4\}$
The orbits of $f^1 = \{1, 2, 4, 3\}$
Are the orbits of $f^0$ and $f^1$ in the same equivalence class or cell? When are two or more different permutations orbits in the same equivalence class?
 A: For a permutation $f$, i.e. a bijective map $f\colon A\to A$, the notation $f^n$ denotes the $n$-fold composition of $f$ with itself. For example $f^3$ is the permutation that maps $a\in A$ to $f(f(f(a)))$ and $f^{-2}$ is the permutation that maps $a\in A$ to $f^{-1}(f^{-1}(a))$, where $f^{-1}$ is the inverse of the bijective map $f$. To find the orbits of a permutation $f\colon A\to A$, look at some $a\in A$ and apply $f$ until you get back to $a$, all the elements you encounter in between are in the orbit of $a$. (If $A$ is infinite, you might never come back to $a$, then the orbit of $a\in A$ really needs to be described as $\{f^k(a)\mid k\in\mathbb Z\}$)
You are asking the wrong questions:

Are the orbits of $f^0$ and $f^1$ in the same equivalence class or cell? When are two or more different permutations orbits in the same equivalence class?

The equivalence classes here consist of elements of $A$, not of orbits. The equivalence classes are the orbits, or cells!
Two elements $a, b\in A$ are in the same orbit of a permutation $f$, or same equivalence classe, or same cell if and only if $f^k(a)=b$ for some $k\in\mathbb Z$.
