In general, symmetry involves mirror-imaging: items are symmetrical if their parts are uniformly balanced and spaced on each side of a dividing line. Items which are not symmetric are asymmetric. In gardening, symmetry implies formality, and asymmetry implies informality.
Formal vs Informal
Strict formality requires control, order, and symmetry: straight lines, perfect balance, and repetition of geometric patterns. Strictly speaking, informality entails an asymmetric variety of freeform shapes in a random order. Most gardeners try to balance the extremes by blending some formal characteristics with a few informal ones. Thus you may have sharp lines and nearly symmetrical arrangements with some variety and contrast thrown in to make the garden comfortable and pleasing.
One general rule to be aware of: formal styles are well-suited close to a house, and informal styles work well farther from the house. The formality nearby helps relate the rigid lines of the house to the surrounding landscape.
Note: One concept utilized in garden design is “balance”. While symmetrical designs are inherently balanced, asymmetrical designs can be balanced as well, provided the “weight”of the design is spread out proportionately. A well balanced design can help create a sense of peace and tranquility in a landscape.