$ \{x : P(x)\} $ vs. $ \{P(x) : x\} $ ---- When are these set-builder notations the same and different? I should clarify that I'm asking for intuition or informal explanations. I'm starting math and never took set theory so far, thence I'm not asking about formal set theory or an abstract hard answer.  
From Gary Chartrand page 216 Mathematical Proofs - 
$\begin{align} \text{ range of } f & = \{f(x) : x \in domf\} = \{b : (a, b) \in f \} \\ 
& = 
\{b ∈ B : b \text{ is an image under $f$ of some element of } A\} \end{align}$
Wikipedia - $\begin{align}\quad \{\text{odd numbers}\} & = \{n \in \mathbb{N} \; :  \; \exists k \in \mathbb{N} \; : \; n = 2k+1   \}     \\
& = \{2n + 1 :n \in \mathbb{Z}\} \end{align}$
But  Why $G/G = \{gG : g \in G \} \quad ? \quad$ And not $\{g \in G : gG\} ?$
EDIT @Hurkyl 10/5.  Lots of detail please.
Question 1. Hurkyl wrote $\{\text{odd numbers}\}$ in two ways.
But can you always rewrite $\color{green}{\{ \, x \in S: P(x) \,\}}$ with $x \in S$ on the right of the colon? How?
$ \{ \, x \in S: P(x) \,\} =  \{ \, \color{red}{\text{ What has to go here}}  : x \in S \, \} $? Is $ \color{red}{\text{ What has to go here}} $ unique?
Qusetion 2. Axiom of replacement --- Why $\{ f(x) \mid x \in S \}$ ? NOT $\color{green}{\{ \; x \in S \mid f(x) \;
 \}}$ ?
@HTFB. Can you please simplify your answer? I don't know what are ZF, extensionality, Fraenkel's, many-one, class function, Cantor's arithmetic of infinities, and the like. 

The first set is the collection of all the $x$ which are both elements in $S$ and satisfy the property $P$.
The second set is the collection of the objects "$P(x)$" for all $x\in S$, for example if $P(x)$ is the function $x^2$ and $S=\Bbb N$ then the result is the set of squares.
