Enriched category theory is the category theory of enriched categories.
Pretty much every notion of ordinary category theory has its analog in enriched category theory, and, by taking the enriching category to be Set, enriched category theory subsumes all of the theory of locally small categories.
Enriched categories have a multitude of uses and applications, that makes studying their general theory quite worthwhile.
Enriched categories may be used to model objects in higher category theory: if the objects of the enriching category $V$ behave themselves as (n,r)-categories, then a $V$-enriched category behaves like a (possibly special) $(n+1,r+1)$-category.
For instance
a Cat-enriched category is a strict 2-category;
since a simplicial set that is a Kan complex is a model for an ∞-groupoid, a Kan complex-enriched category is a model for an (∞,1)-category;
since a simplicial set that is a quasi-category is a model for an (∞,1)-category, a quasi-category-enriched category is a model for an (∞,2)-category.
In practice it is often useful to handle enriched categories such that only certain enriched full subcategories of them are the the desired models for objects in higher category theory. For instance often it is useful to work with general sSet-categories and have a prescription for how to find a Kan complex-enriched full subcategory inside them.
This is achieved notably by combining enriched category theory with model category theory:
an enriched model category or more generally an enriched homotopical category is an enriched category with extra information on how it behaves as a model in higher category. Notably sSet-model categories serve as models for (∞,1)-category theory.
The standard monograph on enriched category is
Also
Francis Borceux, I. Stubbe, Short introduction to enriched categories (pdf)
Emily Riehl, chapter 3 Basics of enriched category theory in Categorical Homotopy Theory
For more references see at enriched category.