There are numerous scientific and engineering fields where the surface dielectric barrier discharge driven by nanosecond pulses (NS-SDBD) has important applications. To improve its performance, more research is still needed on the effects of electrical parameters on the NS-SDBD actuator’s discharge characteristics. In this study, a two-dimensional numerical model based on 13 discharge particle chemical processes was constructed using a numerical simulation approach, producing findings for the NS-SDBD actuator’s voltage–current (V-A) characteristics, discharge profile, and spectrum analysis. Additionally, a comprehensive investigation into the trends and underlying mechanisms of the effects of the voltage amplitude, pulse width, rise time, and fall time parameters on the discharge behavior of the NS-SDBD actuator was carried out. The results show that higher voltage amplitudes increase the maximum current and electron density, which enhances the plasma excitation effect. The peak power deposition during the second discharge is also raised by longer pulse widths and rise times, whereas the total power deposition during the second discharge is decreased by longer fall times.