Meaning of Parallax setting Terraria?

Answer 1

It's an animation technique where parts of an animation's environment are put on different layers which move at different speeds as the animation is played (Or in this case, as your character moves around on and in the World), to simulate the real-life Parallax effect (Notice how things up close appear to move faster than things far away when you move your head from side-to-side?).

At least, that's what I've gathered from a few minutes on Wikipedia.

 

Alternatively, try setting Parallax to 0, run around a cave, then set it to 100 and run around a cave.  See what's different.