In the context of model categories, the left (reps. right) derived functors are a generalization of the traditional notion of a derived functor. The left (resp. right) derived functor $LF$ of a functor $F\colon \mathbf C\to \mathbf D$ where $\mathbf C$ is a model category is exactly a right (resp. left) Kan extension along the localization $\gamma\colon \mathbf C\to\operatorname{Ho}(\mathbf C)$. This terminology of derived functors stems from what happens in the category of chain complexes. In general, computing $LF(X)$ is computing $F(QX)$ where $QX$ is a cofibrant replacement of $X$ where we view an $R$-module $X$ as a chain complex concentrated in degree 0 (the definition of $LF$ is unique up to natural isomorphism). In the category of chain complexes, $QX$ is a projective resolution of $X$, and hence the homology of $LF(X)$ for a right exact functor $F$ is the left derived functors of $F$. For example, if $F$ is $-\otimes_R A$, then the homology of $LF$ will be $\operatorname{Tor}_*^R(-,A)$.
We have a similar story for right derived functors (which are left Kan extensions along the localization functor). Here, $RF(X)=F(RX)$ where $RX$ is a fibrant replacement. For chain complexes, this is an injective resolution, and so the cohomology of $RF(X)$ is the right derived functors of $F$.
If you are unfamiliar with model categories, I'd recommend Homotopy theories and model categories by Dwyer and Spalinski. In particular, example 9.6 deals with what I just outlined.