Is it true that a dihedral group is nonabelian?

Yes, the dihedral groups $D_n$ are nonabelian for $n\ge 3$. It is generated by a rotation $r$ with $r^n=1$ and a reflection $s$ with $s^2=1$. However, you can easily check that a rotation and a reflection will not commute in general. We have $sr=r^{-1}s$ instead for $D_n$ with this presentation.

$D_3$, i.e, the dihedral group of a triangle is isomorphic to $S_3$ which is non-abelian. It can be shown that this is true for $n \geq 3$.

Remark: Some people denote the dihedral group by $D_{2n}$ which is based on the fact the order is $2n$, while some people denote it by $D_n$.

The dihedral groups for $n=1$ and $n=2$ are abelian; for $n\geq 3$, the dihedral groups are nonabelian (this is mentioned on Wikipedia).