The hom-object Hom(A, B) in an enriched category over V is the object in the monoidal category V that plays the role of “morphisms from A to B.” In an ordinary category, hom-objects are sets (the hom-sets); in an enriched category, they can be topological spaces, chain complexes, simplicial sets, or any other type of object in V.
Composition in an enriched category is a map Hom(B, C) ⊗ Hom(A, B) → Hom(A, C) using the tensor product of V, and identity morphisms are maps I → Hom(A, A) from the unit object. The structure on hom-objects is what makes enrichment meaningful: in a topologically enriched category, composition is continuous; in a simplicially enriched category, morphisms form a space whose points are morphisms and whose higher simplices witness homotopies between them.
The internal hom (or exponential object) in a monoidal category is a hom-object that is right adjoint to the tensor product: morphisms A ⊗ B → C correspond bijectively to morphisms A → [B, C]. This adjunction — the closed structure — is the categorical version of the residuation law in a Heyting algebra: a ∧ b ≤ c if and only if a ≤ (b → c).