Monoids, groups, rings, fields, etc. are all very relevant to cryptography. In general algebraic constructions come up more often than you might think. The idea of a homomorphism though not specific to algebra is prevalent in almost any formal domain. Compilers by and large can be considered homomorphisms that preserve certain semantic properties which if you dig deep enough can be expressed as algebraic structures: http://www.logicmatters.net/resources/pdfs/Galois.pdf.

Developing an intuition for all those things is best accomplished by studying abstract algebra.