Definition (triangulated subcategory)

A triangulated subcategory $A$ of a triangulated category $B$ is a nonempty subcategory closed under suspension of objects and such that for all objects $X,Y$ in $A$ if $X\to Y\to Z\to X[1]$ is a distinguished triangle in $B$, then $Z$ is in $A$.

A triangulated subcategory $A$ is called thick if with any object in $A$ it contains all its direct summands in $B$.

Definition (Verdier quotient)

For $A \subset B$ a triangulated subcategory, the Verdier quotient? category $B/A$ which is a triangulated category equipped with a canonical functor $Q^*:B\to B/A$ that is also triangulated (additive and preserving distinguished triangles) and universal among all triangulated functors $B\to D$ which send objects of $A$ to objects isomorphic to $0$.

Definition (Bousfield localization)

Given a thick subcategory $A\subset B$, we say that the Bousfield localization exists if the Verdier quotient functor $Q^*$ has a right adjoint functor $Q_*$ which is then (automatically) triangulated and fully faithful.