Algebras and $\sigma$-algebras are just Boolean algebras. In set theory there are applications (and study) of Boolean algebras, although often we discuss complete algebras. Namely, the union and intersection of any collection, not just finite or countable collections (e.g. power sets).
In measure theory, and its "applicative branch" - probability theory, one uses $\sigma$-algebras to describe the collections of measurable events.
One can then take one step further into set theory where the study of Borel sets and projective sets (both $\sigma$-algebras) is important, and takes quite some of the attention.
Lastly, the theory of Boolean algebras have merits on its own and some surprising applications to other fields of mathematics. Reconstruction theorems of Rubin are such example (given two spaces that an automorphism group of both is isomorphic under some condition, can we deduce the spaces were isomorphic to begin with? Yes, sometimes).