Usage of triangulated categories As the title indicates the question has to do with the usage of triangulated categories. What's the aim behind their study and where do we need them? Is there some algebraic insight behind their study, or some geometric intuition? Because there is a huge theory behind them I guess that there should be a well-defined reason for their existence. 
Applications/Examples are more than welcomed!  
References instead are welcomed too.
 A: 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.
A: 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.
A: Thank you both for your answer, I did upvote them both however wasn't exactly what I was looking for to be honest. For instance, inside this paper here [1] 
the author is exploiting the triangulated structure to derive geometric results. That is the triangulated structure helps to classify certain triangulated subcategories of the derived category $\mathbf{D}(R)$ of a commutative Noetherian ring $R$, and then establishes an one-to-one correspondence between certain subcategories called localized (which are nothing else but thick, i.e. triangulated subcategories closed under taking direct summands, closed under direct sums) and the subspaces of the affine scheme $Spec(R)$.
Also I'm sure that more profound reasons might exist towards the algebro-geometric approach (like in sheaf theory for instance) that justify the reason that such a structure is really needed in our life.
