I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition
of a $\mathbf{S}$-enriched model category, where $\mathbf{S}$ is a monoidal model category. But in the book model structures are introduced only on $\mathsf{Set}$-enriched categories. 



> 
> So, what does a model structure on a $\mathbf{S}$-enriched category mean? 
> 
> 
> 


Is it supposed to be that $\mathbf{S}$ obtains a forgetful functor to $\mathsf{Set}$, and the model structure is defined on the category with respect to the $\mathsf{Set}$ enrichment, or is it something else?



Definition A.3.1.5 of Higher Topos Theory
Your guess is correct, indeed. In general, given any monoidal category $(\mathbf V, \otimes, 1)$, and any $\mathbf V$-enriched category $\mathbf C$, one can always consider the *underlying category* $\mathbf C\_0$ as the ($\mathbf{Set}$-)category having as objects the same objects as $\mathbf C$, and as hom-sets
$$
\mathbf C\_0(x,y):= \mathbf V(1,\mathbf{Hom}\_{\mathbf C}(x,y))
$$
You can easily work out how to define composition, after checking that $\mathbf V(1,-)$ is a lax monoidal functor, as pointed out by Denis in the comments.


Now, if $\mathbf S$ is a monoidal model category, and $\mathbf A$ is a $\mathbf S$-enriched category, "equipping $\mathbf A$ with a model structure" just means "equipping $\mathbf A\_0$ with a model structure", whereas to talk about a *$\mathbf S$-enriched model category* one requires the two extra conditions spelled out in HTT A.3.1.5.


