Let $(\mathcal{X},d,\mu)$ be a non-homogeneous metric measure space satisfying both the upper doubling and the geometrically doubling conditions.  In this paper, we study weighted inequalities of the Calderon-Zygmund operator on $(\mathcal{X},d,\mu)$. Specifically, for $1 <p< \infty$, we identify sufficient conditions for the weight on one side, which guarantee the existence of another weight in the other side, so that the weighted $L^p$ inequality holds.  We deal with this problem by developing a vector-valued theory for Calderon-Zygmund operators on the non-homogeneous metric measure spaces which is interesting in its own right.