In other words, symmetry is the fact that if you ''make some change'' (either in the object itself or in your perspective) the object looks exactly the same. The ''extent'' of symmetry can now be described by the number of such different perspectives you can use. For instance, an equilateral triangle possesses ''some'' symmetry: if you rotate by certain angles, it doesn't change. But rotating by an arbitrary angle does ''not'' send the equilateral triangle to itself. So the equilateral triangle isn't quite as symmetric as the circle.
Symmetries of an object is measured by the set of transformations that map the object to itself. Object can be replaced by a structure, or rule. We're used to thinking of symmetries of concrete objects (like mice, clocks, and historical monuments). But
in physics, we're interested in the symmetry and invariance properties enjoyed by ''laws''. In chemistry, we're interested in the symmetries of small things like molecules.
===Symmetry from a fairness perspective===
This also relates to another important fact: equality, and fairness, translate in group theory to invariance under a certain group action. The group action is the one that permutes the elements.