Consider the definition of existence internal homs for a general monoidal category category $\cal{C}$, mainly the existence of an adjoint for the functor
$$
X \otimes -: \cal{C} \to \cal{C},
$$ 
for each object $X$ in $\cal{C}$.


Denoting this functor by 
$$
hom\_X(-):\cal{C} \to \cal{C}
$$
it is tempting to ask if the functor 
$$
hom: {\cal C} \times {\cal C} \to {\cal C}, ~~~~~~~ (X,Y) \mapsto hom\_X(Y),
$$
gives an enrichment of $\cal{C}$ over itself. Is this correct?
Moreover, is the existence of an enrichment of $\cal{C}$ over itself equivalent to the existence of internal homs?


More generally, when people speak of **internal homs** for a category, not necessary monoidal, are they just talking about an enrichment of the category over itself? Is this what usually understood by "a category with internal homs"?



Enrichments vs Internal homs
As Mike says, it is indeed the case that a closed monoidal structure gives rise to a self-enrichment of a category. This is an often-used fact.


Here's an example of "wrong-way" self-enrichment: the category $\mathsf{Cat}$ of small categories is enriched in itself in the usual way since it is cartesian closed. But it also has another self-enrichment where you take the maximal subgroupoid of each hom-category. This sort of variant enrichment is important e.g. to make $\mathsf{Cat}$ into a simplicial model category.


For that matter, $\mathsf{Cat}$ also has a self-enrichment where you take the usual $\mathsf{Set}$-enrichment, and regard the hom-sets as discrete categories.


