What are applications of rings & groups?

I am following a course in basic algebra, and we have covered rings & groups in class, but I am having trouble visualising them. Are there applications of group &/or ring theory that can be more easily visualized than the abstract object? For instance, are there objects, or properties of objects, that behave as elements of a group in physics, chemistry, or other fields?

Group theory may be viewed roughly as a general study of symmetry. In chemistry this applies to crystals via the study of crystallographic groups, and in art via wallpaper groups. For an example in physics, the Lie symmetry groups of partial differential equations play fundamental roles, e.g, governing conservation laws and separation of variables. See for example Weyl's Symmetry and Budden's Fascination of Groups.

Dihedral groups arise frequently in art and nature. Many of the decorative designs used on floor coverings, pottery, and buildings have one of the dihedral groups as a group of symmetry. Corporation logos are rich sources of dihedral symmetry. Chrysler’s logo has D5 as a symmetry group, and that of Mercedes-Benz has D3. The ubiquitous five-pointed star has symmetry group D5. The phylum Echinodermata contains many sea animals (such as starfish, sea cucumbers, feather stars, and sand dollars) that exhibit patterns with D5 symmetry. Chemists classify molecules according to their symmetry. Moreover, symmetry considerations are applied in orbital calculations, in determining energy levels of atoms and molecules, and in the study of molecular vibrations.

Source : Contemporary Abstract Algebra, Gallian, Chapter 2