The key plasma parameters under different discharge modes, e.g., the heavy-particle and electron temperatures, the electron number density, the non-equilibrium volume of plasmas, play important roles in various applications of gas discharge plasmas. In this paper, a self-consistent two-dimensional non-equilibrium fluid model coupled with an external circuit model is established to reveal the mechanisms related to the discharge modes including the normal glow, abnormal glow, arc, and the glow-to-arc transition mode with the atmospheric-pressure direct-current (DC) argon discharge as a model plasma system. The modeling results show that under different discharge modes, the most significant difference between the preceding four discharge modes lies in the current and energy transfer processes on the cathode side. On the one hand, the current to the cathode surface is mainly delivered by the ions coming from the plasma column under the glow discharge mode due to the low temperature of the solid cathode, while the thermionic and secondary electrons emitted from the hot cathode surface play a very important role under the arc mode with a higher cathode surface temperature and higher ion flux towards the cathode. On the other hand, the energy transfer channel on the cathode side changes from mainly heating the solid cathode under the glow mode to simultaneously heating both the solid cathode and the plasma column under the arc mode with the increase of the discharge current. Consequently, the power density in the cathode sheath (Pc
) is used as a key parameter for judging different discharge modes. And the range of (0.28 ~ 1.2)×1012 W·m-3 is determined as a critical window of Pc
corresponding to the glow-to-arc mode transition for the atmospheric-pressure DC argon discharges, which is also verified well by comparing with the experimental results in this study and the data in the previous literatures.