A simplicially enriched category is a category enriched over the cartesian monoidal category sSet of simplicial sets.
These enriched categories are sometimes, somewhat imprecisely, called just simplicial categories.
There is a related notion of simplicial groupoid with the added requirement that all arrows in the categories concerned are isomorphisms.
Since simplicial sets that are Kan complexes are an incarnation of ∞-groupoids, an -category all whose hom-objects happen to be Kan complexes may be regarded as a category enriched in ∞-groupoids. By the logic of (n,r)-category theory this should be a model for an (∞,1)-category.
Treating simplicial categories tis way as models for -categories is one of the central tools in homotopy coherent category theory.
Indeed, there is a model structure on simplicial categories whose fibrant objects are Kan-complex enriched categories, and which is one model for the (∞,1)-category of (∞,1)-categories.
By a web of Quillen equivalences this is related to the other incarnations of -categories. Notably to quasi-categories and complete Segal spaces. For more on this see