The characteristic of a structure (usually a molecule) that makes it impossible to superimpose it on its mirror image. Also called handedness.
The American Heritage® Science Dictionary.
Copyright © 2002. Published by Houghton Mifflin. All rights reserved.
Informally, you can get away with just using "handedness", because "chirality" is just Lord Kelvin's Hellenized version of the English word. This comes from the fact that most people's left hands are rough mirror images of the corresponding right hands.
When you're introduced to vector spaces, you'll learn about the "right-hand rule" which determines the direction of a cross product.
You might enjoy Martin Gardner's book The New Ambidextrous Universe, which explores many aspects of symmetry and charality/handedness.
You've asked what happens to an "achiral" object like a square, but you've inadvertently asked about two different systems. An "unlabeled" square is achiral and changes nothing when it's reflected, because we don't care which vertex is which. But if you go and do something such as label the vertices, then you've introduced chirality to the system.
Essentially, you've taken an undirected graph and turned it into a directed one. Draw a square on a piece of paper, then label its vertices clockwise. Then, reflect the graph in a mirror (or look through the other side of the paper). The labels you see will be counterclockwise.