The theory of universal algebras was well-developed in the twentieth century. [It] provides a basis for model theory, and [provides] an abstract understanding of familiar principles of induction, recursion, and freeness.
The theory of coalgebras is considerably [less] developed. Coalgebras arise naturally, as Kripke models for modal logic, as automata and objects for object oriented programming languages in computer science, and more.
Jesse Hughes, A Study of Categories of Algebras and Coalgebras