TITLE: What is the importance of “variety of algebras” in Universal Algebra?
QUESTION [12 upvotes]: Given an algebraic category, Birkhoff's Variety Theorem gives a categorical characterization of the full subcategories whose object-class forms a variety (i.e. can be defined by equations in the sense of Model Theory).
The theorem is often stated as being of fundamental importance to Universal Algebra. As far as its importance for metamathematical questions is concerned, this does not surprise me, as it describes a connection between Model Theory and Universal Algebra. But what about its “internal” importance for Universal Algebra? Suppose we are studying a certain class of objects in some algebraic category. In how far could it be useful to know whether this class forms a variety?
My question only concerns the one implication of Birkhoff's theorem of course. It is clear that the result that varieties are closed under the taking of products, subalgebras and homomorphic images has a wide range of possible applications. But what about the converse?

REPLY [4 votes]: Reformulating part of Alex Kruckman's answer: Knowing that some class $\mathcal{A}$ of algebras has an equational presentation is useful because we then know that an algebra belongs to $\mathcal{A}$ if every finitely generated subalgebra does.