When working modulo a given number, a Quadratic Residue is a square, modulo that number. In other words, when you square something and reduce module m, what's left, the residue, is the quadratic residue.

These turn up more often than you might expect in number theory, so there is a wide range of notation devoted to this and related concepts. The Jacobi Symbol is one such idea that seems pointless and unmotivated when first studied, but eventually is seen to be an indispensible concept.