A homomorphism is a function between algebraic structures that preserves the operations. If (A, ∗) and (B, ·) are magmas, a homomorphism f: A → B satisfies f(a ∗ b) = f(a) · f(b) for all a, b ∈ A. For richer structures (monoids, groups, rings, lattices), the homomorphism must preserve all the operations and distinguished elements.
A homomorphism that is also a bijection is an isomorphism — it establishes that two structures are algebraically identical. A homomorphism from a structure to itself is an endomorphism; a bijective endomorphism is an automorphism. The kernel of a homomorphism (elements mapped to the identity) encodes what information the homomorphism discards, and the first isomorphism theorem says the image is isomorphic to the quotient by the kernel.
Homomorphisms are the morphisms of algebraic categories: the category of groups has groups as objects and group homomorphisms as morphisms; the category of rings has ring homomorphisms; and so on. A functor between categories is a higher-level homomorphism — it preserves composition and identity morphisms, just as a homomorphism preserves the binary operation and identity element.