$[b,T]$ denotes the commutator of generalized Calderón-Zygmund operators $T$ with Lipschitz function $b$, where $b ∈ \rm{Lip}_β(R^n)$, $(0<β≤1)$ and $T$ is a $θ(t)$−type Calderón-Zygmund operator. The commutator $[b,T]$ generated by $b$ and $T$ is defined by
$$[b,T] f(x)=b(x)Tf(x)−T(bf)(x)=∫k(x,y)(b(x)−b(y))f(y)dy.$$
In this paper, the authors discuss the boundedness of the commutator $[b,T]$ on weighted Hardy spaces and weighted Herz type Hardy spaces and prove that $[b,T]$ is bounded from $H^p(ω^p)$ to $L^q(ω^q)$, and from $H\dot{K}^{α,p}_{q_1}(ω_1,ω^{q_1}_2)$ to $\dot{K}^{α,p}_{q_2}(ω_1,ω^{q_2}_2)$. The results extend and generalize the well-known ones in [7].