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.