If $P$ is the transition matrix belonging to a markov chain, then what does it mean that $P^n$ is irreducible for every $n\in\mathbb{N}$?
For $n=1$ it means that all states communicate with each other, i.e. for all states $i,j$ it is $$ \mathbb{P}(\exists m\in\mathbb{N}: X_m=j|X_0=i)>0. $$
But what does it mean for $n\geq 2$?
Edit
Does it maybe mean that for any states $i,j\in E$ it is
$$ \mathbb{P}(\exists m\in\mathbb{N}: X_{n\cdot m}=j|X_0=i)>0? $$
It means that all states make up a single subclass. In other words, it means that matrix $P$ is primitive (as also noted in comments by @Did it's aperiodic which I believe is another term for that).
This can easily be seen from the following theorem:
Theorem. (Gantmacher F.R. The Theory of Matrices, 1960, Vol. 2, P. 81, Theorem 9.) If $A \geqslant 0$ is an irreducible matrix and some power $A^q$ of $A$ is reducible, then $A^q$ is completely reducible, i.e., $A^q$ can be represented by means of a permutation in the form $A^q = \operatorname{diag} \{ A_1, \dots, A_d \}$, where $A_1, \dots, A_d$ are irreducible matrices having one and the same maximal characteristic value. Here $d$ is the greatest common divisor of $q$ and $h$, where $h$ is the index of imprimitivity of $A$.
Since $d$ is $1$ for any $q$, $h = 1$. QED.