The practitioners of algebraic combinatorics are interested in combinatorial aspects of algebraic objects in order to explain, make concrete, and describe their properties. Sometimes the roles are reversed, with algebraic machinery providing insight into traditionally combinatorial domains.
A family of algebraic structures that some of us give a lot of attention is that of Coxeter groups. They appear, for example, in the study of symmetries; as special cases of Coxeter groups one finds the symmetries of the five platonic solids (as well as higher-dimensional regular polytopes). Other examples include the Weyl groups associated with root systems that are important in Lie theory and, consequently, in theoretical physics.