If $J_i$ represent the angular momentum operators, then a scalar operator $S$ (rank-0 tensor) is defined as an operator which satisfies $$[S,J_i]=0$$ for $i=1,2,3$.
$A_i$ is a vector (rank-1 tensor) operator, if it satisfies $$[J_i,A_j]=i\hbar\epsilon_{ijk}A_k$$
How does a rank-2 tensor operator defined in terms of commutators?
Given \begin{align} \hat L_\pm \vert \ell m\rangle &= \sqrt{(\ell\mp m)(\ell\pm m+1)}\vert \ell,m\pm 1\rangle\, ,\\ \hat L_0 \vert \ell m\rangle &= m \vert \ell m\rangle \end{align} then by definition $\hat T^{(\ell)}_m$ commutes as \begin{align} [\hat L_\pm, \hat T^{(\ell)}_m]&=\sqrt{(\ell\mp m)(\ell\pm m+1)}\,\hat T^{(\ell)}_{m\pm 1}\, ,\\ [\hat L_0, \hat T^{(\ell)}_m]&=m\,\hat T^{(\ell)}_{m}\, , \end{align} valid for any $\ell,m$.
