According to Wikipedia,
An irreducible $n$-manifold, is one in which any embedded $(n − 1)$-sphere bounds an embedded $n$-ball.
What I understand from this definition is,
if $\Bbb S^{n-1}$ is a embedded submanifold of $M$ then there is a $n$-ball such that $\partial B_n=\Bbb S^{n-1}$.
is this correct? I don't know why I think this happen always. i.e. such $B_n$ exists always. Is there any $2$-dimensional example of irreducible $2$-manifold for better understanding?
For the sake of providing this with an answer, there are various counter examples. The punctured plane is one, as Wojowu says in the comments.
We also have $\mathbb{R}^3 \backslash \{0\}$, and other similar cases. $T^2$ is another good example for a $2$-dimensional case.