equivalences in/of $(\infty,1)$-categories
As for localization of ordinary categories, there are slightly different notions of what a localization of an (∞,1)-category is.
One definition is in terms of reflective (∞,1)-subcategories:
A localization , in this sense, of an (∞,1)-category $C$ is a functor $L : C \to C_0$ to an $(\infty,1)$-subcategory $C_0 \hookrightarrow C$ such that with $c$ any object there is a morphism connecting it to its localization
in a suitable way. This “suitable way” just says that $f$ is left adjoint to the fully faithful inclusion functor.
Since localizations are entirely determined by which morphisms in $C$ are sent to equivalences in $C_0$, they can be thought of as sending $C$ to the result of “inverting” all these morphisms, a process familiar from forming the homotopy category of a homotopical category.
An (∞,1)-functor $L : C \to C_0$ is called a localization of the (∞,1)-category $C$ if it has a right adjoint (∞,1)-functor $i : C_0 \hookrightarrow C$ that is full and faithful.
In other words: $L$ is a localization if it is the reflector of a reflective (∞,1)-subcategory $C_0 \hookrightarrow C$.
This is HTT, def. 5.2.7.2.
Localizations of $(\infty,1)$-categories are modeled by the notion of left Bousfield localization of model categories.
One precise statement is: localizations of (∞,1)-category of (∞,1)-presheaves $C = PSh_{(\infty,1)}(K)$ are presented by the left Bousfield localizations of the global projective model structure on simplicial presheaves on the simplicial category incarnation of $K$.
∞-stackification (or (∞,1)-sheafification) is the localization of an (∞,1)-category of (∞,1)-presheaves to the $(\infty,1)$-subcategory of (∞,1)-sheaves.
This is the topic of section 5.2.7 and 5.5.4 of