The following image is from page 1019 of http://www-fourier.ujf-grenoble.fr/~lanneau/references/masur_tabachnikov_chap13.pdf
From what I understand about unfolding billiards we are representing the path of a billiard in the original quadrilateral with a line of constant slope (the original slope of the billiard). In the third polygon (from the left) the billiard is hitting a side and looks as if its next move will be to hit the line with slope one. However, in the next polygon the billiard is headed back into the vertical side. Isn't this path supposed to reflect the path of the billiard in the original quadrilateral?
Is there a nice way to determine how to reflect the polygon at any step of the unfolding?
To explicate the general procedure @Ross described, I single it out once more:
You need to reflect the table in the side being hit.
Associated intuition is as if you're looking at the ball being reflected against a mirror (after the ball hits the mirror, you start looking at it through the mirror). This means that the third and fifth reflections in the included image are done wrong.
Once you get a bit of intuition for this kind of thing it will all make sense. You can experiment with a piece of paper and some simpler shapes (e.g. an equilateral triangle or a rectangle). Bonus of this is that it's actually fun to do :).