A $p$-group is a group where the order of every group element is a power of the prime $p$. The presentation of a dihedral group $D_n$ of order $2 n$ is as follows.
$$D_n = \langle x, y \mid x^n = y^2 = (xy)^2 = e, y x=y^{-1}x\rangle$$
So, $D_n$ can be a $p$-group only when $n = 2^m$, when $m \in \mathbb{Z}$ ,because that is when $x^{2^m} = y^{2^1} = e$ for $p = 2$.
Am I getting it right?