I've always found this concept difficult to grasp, so I'm going to use a very particular (cumulative) approach here that'll hopefully help me understand them purely category-theoretically.
Every object $X \in \mathcal C$ induces a constant functor $\Delta_{\mathcal J}(X)\colon \mathcal J \to \mathcal C$ which sends every object in $\mathcal J$ to $X$ and every morphism to $1_X$.
Remark. $\Delta_{\mathcal J}\colon \mathcal C \to {\mathcal C}^{\mathcal J}$ is itself a functor called the diagonal functor.
The limit (a.k.a. projective limit, inverse limit) of a functor $F\colon \mathcal J \to \mathcal C$, $\underset \longleftarrow \lim F$ is a natural transformation $\pi\colon \Delta_{\mathcal J}(L) \Rightarrow F$ (a cone) such that for every other natural transformation $p\colon \Delta_{\mathcal J}(C) \Rightarrow F$, there exists a unique morphism $!\colon C \to L$ such that $\pi_X$, $p_X$, and $!$ commute for all $X$.
In other words, $L$ is a terminal object in the category of cones.
Generally one speaks of extending a functor $F\colon \mathcal C \to \mathcal E$ along another functor $K\colon \mathcal C \to \mathcal D$, to produce a functor with a universal property, $\mathrm{Ran}_K F\colon \mathcal D \to \mathcal E$ the right Kan extension. So in a sense we're "moving" the domain of $F$ to the codomain of $K$, or making up a functor that tries to "finish the job" that $K$ does by modeling $F$.
The universal property is that there is a universal natural transformation $\epsilon_X\colon \mathrm{Ran}_K F(K(X)) \to F(X)$ such that
for any other candidate $G\colon \mathcal D \to \mathcal E$ with a natural transformation $\delta_X\colon G(K(X)) \to F(X)$,there exists a unique $!_X\colon G(X) \to \mathrm{Ran}_K F(X)$ such that $\delta$ commutes with $\epsilon$ after $!$.
Look familiar?
Let's define the point category ($\bullet$) as a category with a single object and morphism.
A limit is a special case of a right Kan extension ($\epsilon_X \simeq \underset \longleftarrow \lim F$) when, you guessed it, $\mathcal D = \bullet$. In this case, $K$ has no choice but to be a constant functor $\Delta_{\mathcal J}(\bullet)$ for the only object in $\mathcal D$, and $\mathrm{Ran}_K F$ and $G$ designate singular objects in $\mathcal E$.
By abuse of notation, it might look something like this:
Let's say you have a right adjoint $G\colon \mathcal D \to \mathcal C$. You can get the left adjoint using $\mathrm{Ran}_G 1_D$. In $\mathrm{Cat}$, this looks like
The natural transformations look like
(I renamed the candidate left adjoint to $T: \mathcal C \to \mathcal D$ here.)
This is only valid if the Kan extension exists and is absolute, meaning it's preserved across all functors. I won't go into detail about what preservation means here.