Given two sets with some structure imposed on it - for instance, a relation, or an operation, or an algebra - a homomorphism is a mapping from one to the other that preserves that structure.

E.g., the natural numbers can be mapped to the even natural numbers by the mapping x -> 2x; this is a homomorphism with respect to the usual ordering of numbers.

x -> 2^x is a homomorphism to map addition onto multiplication.

Isomorphisms are the most common.