What does post mean in crochet?

The Word "post" in crochet is referring to the whole stitch--generally a "tall" stitch (such as a double crochet, triple crochet, etc).

The reason the instructions refer to the stitch in the row below, is so the crocheter understands that you are to stitch around the stitch below--the "post" of the stitch. Otherwise, if the instructions stated that you were to double crochet the next stitch, you would understand that you would be double crocheting into the top of the stitch below, versus crochet around the length of the stitch.

You also notice that the instructions will say to "back post double crochet (bpdc)" which means to bring your stitch around the post from the back position. So, "front post, double crochet (fp dc)" would mean to bring your stitch in from the front side of your work.