The three types of equivalence (structural, automorphic, and regular) have progressively less strict definitions of what it means for two actors to be "equivalent". And, as we make the definitions less strict (which is not the same as making them less precise!), we are able to understand social networks at increasing levels of abstraction.
Structural equivalence is the most "concrete" form of equivalence. Two actors are exactly structurally equivalent if they have exactly the same ties to exactly the same other individual actors. Pure structural equivalence can be quite rare in social relations, but approximations to it may not be so rare. In studying a single population, two actors who are approximately structurally equivalent are facing pretty much the same sets of constraints and opportunities. Commonly we would say that two actors who are approximately structurally equivalent are in approximately the same position in a structure.
Automorphic equivalence is a bit more relaxed. Two actors may not be tied to the same others, but if they are embedded in the same way in the larger structure, they are equivalent. With automorphic equivalence, we are searching for classes of actors who are at the same distance from other sets of actors - that is, we are trying to find parallel or substitutable sub-structures (rather than substitutable individual actors).
Regular equivalence deserves special attention because it gets at the idea of the "role" that an actor plays with respect to occupants of other "roles" in a structure. The idea of a social role, which is "institutionalized" by normative and sanctioned relationships to other roles is at the very core of the entire sociological perspective.
The definitions of the forms of equivalence discussed here are quite precise (though my discussion doesn't have much mathematical rigor). The notions of equivalence provide quite rigorous ways of defining and thinking about core analytical tools in sociology - individuals' positions in groups, types of structures, and social roles. This is a huge advance over the sometimes quite imprecise and contradictory verbal treatments found in much of our literature.
But, real world social networks are often quite messy, may not be fully realized (that is, not in equilibrium), and/or may be badly measured. The search for equivalence in real data can be a somewhat complicated matter with a number of vexing choices to be made. We'll spend some time with these practical issues in the next three chapters.