The Kan fibration is similar to the Kan filler condition for Kan complexes. It is the simplicial realization of the Serre fibration.
Take two simplicial sets $X$ and $Y$. Say there's a morphism $p\colon X \Rightarrow Y$.
Also say there's an embedding of horns into $X$ and an embedding of $n$-simplices into $Y$ like this:
$$\begin{CD} \Lambda^n_k @>>> X \\ @VVV @VVpV \\ \Delta^n @>>> Y \end{CD}$$
Such that the diagram commutes
Now exactly what this means is special here because the left and bottom morphisms have a very specific meaning. The left morphism is an inclusion and the bottom morphism is simply a specific $n$-simplex in $Y$. This means that for some horns in $X$ there are accompanying complete simplices in $Y$.
So it can be thought that $p$ has an implicit mapping from certain horns in the simplicial set $X$ to complete simplices in $Y$.
So we'll say that $p$ is a Kan fibration if for every horn in $X$ that has an accompanying completion in $Y$, the completion of that horn exists in $X$.
In other words, a "lift" exists (similar to the one in the Kan filler condition) taking the completion of every one of those horns to $X$, and it maps through $p$ to $Y$ in a way that commutes with the embedding of the completion in $Y$.
So Kan fibrations take some horns to complete simplices that exist in both $X$ and $Y$.
$p$ is an inner (Kan) fibration if there is no lift for any horn $\Lambda^n_0$ or $\Lambda^n_n$.
A simplicial set $X$ is called a Kan complex if the morphism $p!\colon X \Rightarrow \Delta^0$ is a Kan fibration.
$$\begin{CD} \Lambda^n_k @>>> X \\ @VVV @VVp!V \\ \Delta^n @>>!> \Delta^0 \end{CD}$$
(imagine a dotted arrow going from $\Delta^n$ to $X$)
Note that $\Delta^0$ is a terminal object in $\mathrm{sSet}$, so the morphisms marked with $!$ are unique.
Quasicategories can be defined in a similar way using inner fibrations.