Symmetrical balance is usually found in traditional interiors. Symmetrical balance is characterized by the same objects repeated in the same positions on either side of a vertical axis, for example you might remember old rooms where on each side of a room is an exact mirror of the other. This symmetry also reflects the human form, so we are inately comfortable in a balanced setting.
Asymmetrical balance is more appropriate in design in these days. Balance is achieved with some dissimilar objects that have equal visual WEIGHT or eye attraction. Assymetrical balance is more casual and less contrived in feeling, but more difficult to achieve. Asymmetry suggests movement, and leads to more lively interiors.
Radial symmetry is when all the elements of a design are arrayed around a center point. A spiral staircase is also an excellent example of radial balance. Though not often employed in interiors, it can provide an interesting counterpoint if used appropriately.