Symmetry, arguably, is an aesthetic; it measures how nice something looks. But there is a more quantitative aspect to it. Namely, something symmetric is something that ''looks the same'' from different angle. For instance, if your face possesses left-right symmetry, then that means that it looks exactly the same as the face you see in the mirror. If, on the other hand, it didn't possess left-right symmetry (say, if you had a mole on your left cheek) then your mirror image looks ''different'' from your own image. In the mirror image, the mole is on the right cheek.
A circle is an example of a ''lot'' of symmetry. You could rotate the paper any amount around the center of the circle, and you still get a circle.
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.
===Symmetry from a fairness perspective===