(Let me start off by saying that I've yet to meet a mathematician who can clearly and formally define algebraic logic. I'll try anyway...)

As has been noded all over the place (c.f. 1, 2, 3) the field of mathematical logic concerns itself with the relation between semantics (in the form of structures) and syntax (in the form of languages). There are primarily two types of structures that get used on the semantic side relational structures (e.g. Kripke models) and algebras (e.g. Boolean algebras).

Since the study of algebras has a long and rich history many interesting and useful techniques have been developed. Thus by studying the algebraic semantics of a logic new results can be established that might not be easy to derive by just looking at the syntax.

This approach is quite often referred to as algebraic logic and involves trying to generate algebraic semantics for a particular logic and then using the structure of this semantics to delve into the properties of such a logic.

The article "Nineteenth Century Roots of Algebraic Logic" by Anellis and Houser [1988] gives some insight into the development of this field. Important names in the early development of this field include C. S. Pierce and F. Schroder with much of the current developments in this field coming from the early work of A. Tarski on relation algebras.

Related fields in mathematics include mathematical logic, model theory, universal algebra and modal logic.