This post is a work in progress, please feel free to add or embellish...
Most of the details here come from balpha's
post.
Background
There are two basic ways to use the replayer: first, you can use the replayer to simply display a FEN position, and second, you can post a pgn game and then others can click through the moves.
The basic idea is that you post the pgn or FEN in a code block. In order to create a code block, indent the pgn or FEN by four spaces. As a shortcut, you can either click the editor's "Preformatted text" button, or press Ctrl-K.
Position
In order to simply show a position, just post the FEN string in a code block. It will automatically be converted into a diagram.
Example:
r1br2k1/pp3ppp/2n2n2/1Bb3B1/8/2p2N2/PPP2PPP/R4RK1 w - - 1 13
Is converted into:

Entire Game
In order to display an entire game, you can post the entire pgn in a code block. Most chess software allows you to copy/paste the pgn from a game, but wikipedia also has a good summary if you want to just type the pgn yourself.
There is one caveat - the pgn you post must include a FEN header.
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]
This was done in order to create an "opt-in" experience for the replayers (i.e. only display a replayer if you actively choose to have one), but this is open for debate.
As a shortcut, if you are starting from the start position, you can just include a blank FEN header:
[FEN ""]
Additional Features
It is possible to enter entire variations and annotations for your games as well. If you simply copy/paste a pgn file from your external program, it will usually work correctly (after you add a FEN string). In order to enter variations, you can use parentheses for the variation moves. You can enter comments in curly braces too:
1. e4 e5 2. Nf3 Nc6 (2... Nf6 {Nf6 instead of Nc6 is called the Petroff Defense})
Variations can be nested as well.