Usage of triangulated categories

As far as I'm aware, one of the main reasons for the introduction of triangulated categories (in Verdier's thesis) was to be able to define derived categories. One simple motivation for derived categories is that they "correct" the problem that quasi-isomorphisms of (co)chain complexes (that is, (co)chain maps which induce isomorphisms in (co)homology) are not necessarily invertible!

More precisely, if $\mathcal{A}$ is an abelian category and we write $K(\mathcal{A})$ for the category whose objects are (co)chain complexes of objects of $\mathcal{A}$ and whose morphisms are homotopy classes of (co)chain maps, then there is a category $D(\mathcal{A})$ and a functor $P:K(\mathcal{A})\to D(\mathcal{A})$ which turns quasi-isomorphisms in $K(\mathcal{A})$ into isomorphisms in $D(\mathcal{A})$ and is universal with respect to this condition (i.e. any other functor $F:K(\mathcal{A})\to \mathcal{C}$ which turns quasi-isomorphisms into isomorphisms, factors through $P$).

For example, a very common procedure is as follows. Take some object in which you are interested, associate to it (in some manner) a (co)chain complex and then take (co)homology. We then want to investigate how much the (co)homology "knows" about our original object. The derived category is the "right" abstract setting for this type of question.

Triangulated categories were originally invented to allow you to "lift" long exact sequences to the homotopy category of chain complexes. For example, if you have a sufficiently nice pair of topological spaces $(X,A)$, then you have the long exact sequence $$ \cdots \to H_k(A) \to H_k(X) \to H_k(X,A) \to H_{k+1}(A) \to \cdots $$ Then, in the homotopy category $\text{K}(\textbf{Ab})$ we have a morphism of complexes induced from the inclusion $$ C_\bullet(A) \xrightarrow{\Phi} C_\bullet(X) $$ The key construction is then "Cone Construction" which provides something like the cokernel (homotopy cokernel) of $\Phi$ in $\text{K}(\textbf{Ab})$. We write it as $$ C_\bullet(A) \xrightarrow{\Phi} C_\bullet(X) \to \text{Cone}_\bullet(\Phi) $$ The structure of a triangulated category on $\text{K}(\textbf{Ab})$ calls this diagram a distinguished triangle, which we write down slightly differently as $$ C_\bullet(A) \xrightarrow{\Phi} C_\bullet(X) \to \text{Cone}_\bullet(\Phi) \xrightarrow{[+1]} $$ You will then be able to see that the axioms of a triangulated category essentially are describing the behavior of these diagrams called distinguished triangles. What these triangles then let you do is have a long exact sequence before taking cohomology. If you apply the cohomology functor $$ H_0: \text{K}(\textbf{Ab}) \to \textbf{Ab} $$ you will see that you recover the long exact sequence from topology. This follows after a diagram chase.

TL;DR

Triangulated categories add homotopy colimits in $\text{K}(\textbf{Ab})$ by hand.