Symmetry refers to the similarity of elements and their mirroring or repetition around one or more axis. It often is perceived as high-level ordering and, as a result, goes along way towards achieving unity and balance.

Asymmetry is the lack of mirroring of similar around axis points. But it isn't necessarily a bad thing: the imbalance created by an asymmetrical layout can move the reader's eye through the composition.  

In the world of symmetry, there is exact symmetry around a vertical, horizontal, or some other line parallel to the picture plane; there is radial symmetry, where similar objects are rotated about an axis perpendicular to the picture plane (a.k.a. the z axis); and there is approximate symmetry where elements other than shape account for the symmetry (in the example above, the collective weight of the group of shapes approximately equals the weight of the orange hexagon).  

In fact, the whole of the example above exhibits a kind of approximate symmetry. Can you see it?