Birkhoff's Variety Theorem is indeed of fundamental importance to Universal Algebra, notably for the stream of research it has generated, inside and outside of Universal Algebra. Here are a few examples.
The book  is entirely devoted to varieties of groups. See also the paper Varieties of groups by B.H. Neumann. One important result in this field states that the variety of groups generated by a finite group is finitely based. This result does not hold for arbitrary algebras (groupoids and semigroups are counter-examples).
But there is a large literature on the following question:
Is it decidable whether the variety generated by a given finite algebra is finitely based?"
Birkhoff's variety theorem has also been extended in various directions. For instance, it has been extended to ordered algebras in  and to (pseudo)varieties of finite algebras in [1, 6] (the equations are now profinite equations) and to (pseudo)varieties of finite first-order structures in .
These results in turn have found important applications in the study of regular languages through Eilenberg's variety theorem [3, p. 194].
Finally, let me mention the little known but very nice book , which contains some interesting material on varieties and equational theories.
 B. Banaschewski, The Birkhoff theorem for varieties of finite algebras, Algebra Universalis 17 (1983), 360–368.
 S. Bloom, Varieties of ordered algebras, J. Computer System Sciences 13 (1976), 200-212.
 S. Eilenberg, Automata, Languages and Machines, Vol. B, Academic Press, coll. Pure and Applied Mathematics (no 59), 1976, xiii+387 p.
 Hanna Neumann, Varieties of groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37, Springer-Verlag, Berlin, 1967.
 J.-E. Pin and P. Weil, A Reiterman theorem for pseudovarieties of finite first-order structures, Algebra Universalis 35 (1996), 577–595.
 J. Reiterman, The Birkhoff theorem for finite algebras, Algebra Universalis 14 (1982), 1–10.
 Wechler, Universal Algebra for Computer Scientists, EATCS Monographs on Theoretical Computer Science 25 (1992)