Definition
The $n$-step transition probability from state $i$ to state $j$ is
$$ p_{ij}^{(n)}=\mathbb P(X_n=j \mid X_0=i). $$Key identity
The entry $p_{ij}^{(n)}$ is the $(i,j)$ entry of the matrix $P^n$:
$$ P^n = \big(p_{ij}^{(n)}\big). $$The one-step case is
$$ p_{ij}^{(1)} = p_{ij}. $$Consequence
Matrix powers of $P$ describe the distribution of the chain after multiple steps, not just one step.