The high post area is located at the elbows, where the free throw lane and free throw line intersect. The high post is used as a starting point for many offensive plays via screens, passes, or other actions.
The low post is closer to the basket than the high post, down near the block. This is likely what someone is referring to if they mention "the post area."
The low post is similar to the high post in use, but an entire phase of the game revolves around the low post as well.
Players who often position themselves in the low post (or "post up") are called post players, and post players use post moves to score the basketball. These post moves include a series of turns, fakes, and steps that get a post player closer to the basket or create enough space for an open shot.
A player who is frequently used in the low post or who is specific to the post area on a play may be referred to as a "post player" or even shortened to "post."