Recently, origami structures and origami mechanical metamaterials receive extensive attention from the science and engineering communities due to the infinite design space, excellent deformability, extraordinary mechanical properties, and wide application potentials. In particular, some origami structures have been well studied due to their unique bistability. Note that origami structures and origami metamaterials are always composed of multiple cells; however, for multi-cell origami structures, their multistability characteristics and the induced dynamical behaviors have not been well understood. On the basis of the bistable stacked Miura-ori structure, this paper studies an origami structure connected by two heterogeneous cells in series based on force balance. Static analysis suggests that the two-cell series structure have four different stable configurations, exhibiting a multi-stable profile. Dynamical analysis reveals that the two-cell series origami structure presents significantly different natural frequencies at the four stable configurations. With the increase of the excitation amplitude, the multistability of the two-cell series structure could induce complex nonlinear dynamical responses, including intrawell and interwell oscillations that are sub-harmonic, super-harmonic, or even chaotic. They can be classified into nine types based on the response amplitude characteristics. Moreover, the basin of attraction and the basin stability of these dynamical responses are examined. The results indicate that the basin stabilities (i.e., the appearing probabilities) of these types of dynamical response are significantly different and closely relate to the excitation amplitude. In summary, the outcomes of this paper, i.e., the static characteristics of the two-cell series structure, the classification on dynamical responses, and the evolution rule of the basin stabilities with respect to the excitation amplitude, would contribute to the understanding on the nonlinear dynamics of multi-stable origami structures, and provide the basis for controlling the nonlinear dynamical responses.