The unit object I in a monoidal category is the object that acts as an identity for the tensor product: I ⊗ A ≅ A ≅ A ⊗ I for every object A, via coherent isomorphisms. It is the categorical analogue of the identity element in a monoid.
In the category of sets with Cartesian product, the unit object is any singleton set {∗}. In the category of vector spaces with tensor product, the unit is the base field. In the category of abelian groups with tensor product over ℤ, the unit is ℤ itself.
In an enriched category, the identity morphism at each object is a map from the unit object I to the hom-object Hom(A, A). This mirrors how the identity element of a monoid is picked out by a map from the one-element set. The unit object thus provides the “do nothing” morphism in the enriched setting.