I'm not an expert, but, here is my understanding. Right-fibrations are important because they are the infinity-version of a category fibered in groupoids (that is an infinity-category fibered in infinity-groupoids). In particular, given an infinity-category $C$,there is a model structure on $sSet/C$, called the contravariant model structure, such that the fibrant and cofibrant objects are precisely the right-fibrations over $S$, and this model structure is Quillen-equivalent (through a generalization of the Grothendieck construction) to the projective model-structure of simplicial presheaves over $w(C)$, where $w(C)$ is a simplicial category and $w$ is the left-adjoint to the homotopy-coherent nerve. Both of these (simplicial) model categories model $Fun(C^{op},\infty-Gpd)$- the infinity-categeory of "weak presheaves in infinity groupoids". So, the upshot is, right fibrations are the infinity-analogue of Grothendieck fibrations in groupoids and provide a model for weak presheaves. This presheaf infinity-topos is the starting point for higher topos theory; infinity topoi are just left-exact (accessible) localizations of such presheaf-infinity categories.

Now, dually, left-fibrations are a model for "infinity-categories COfibered in infinity-groupoids".

The next step, is Cartesian-fibrations. Cartesian-fibrations are the infinity-version of categories fibered in categories (not necessarily groupoids), i.e. they are "infinity categories fibered in infinity-categories". Nearly everything above goes through again, except we need to work with marked-simplicial sets, where we "mark the cartesian-edges".

Again, dually, CoCartesian fibrations model "infinity categories cofibered in infinity-categories". You may wonder why we need both notions? In fact, we need both notions TOGETHER in order to define adjunctions. An adjunction between two infinity-categories $C$ and $D$ is a functor $K \to \Delta[1]$ which is simultaneously a Cartesian-fibration and a CoCartesian fibration, together with Joyal-equivalences $K_{0} \cong C$ and $K_{1} \cong D$. This definition is a generalization to the infinity-world of a characterization of adjunctions using cographs.

Now, Kan-fibrations are a relic of homotopy theory. They are the fibrations on the Quillen-model structure on simplicial sets. In a similar spirit, categorical fibrations are the fibrations in the Joyal-model structure on simplicial sets. Other than that, they are not that well behaved; they don't really play a role in infinity-category theory.

Finally, inner fibrations, as far as I know, are only used in defining Cartesian fibrations. That is, a Cartesian fibration is defined to be an inner fibration satisfying extra properties.

I hope this helps.