Homomorphisms

morphe means shape in greek, so homomorphism means same shape.

homomorphism is a structure-preserving map between two algebraic structures of the same type. Homomorphisms of vector spaces are also called linear maps

map, i.e relation for $f: S \to T$ that preserves the structure of the algebraic structure so if they both define the same operation *, then the homomorphism will preserve the operation. Preserving the operation means that it doesn't matter if you apply the operation before or after the homomorphism i.e the mapping. Why is this useful think of linear transformations in linear algebra?

Isomorphisms

bijective homomorphism, i.e a homomorphism that is also a bijection can go back and forth between the two structures.