The electronic structures of a MoS₂ monolayer are investigated with the all-electron first principle calculations based on the density functional theory (DFT) and the spin-orbital couplings (SOCs). Our results show that the monolayer MoS₂ is a direct band gap semiconductor with a band gap of 1.8 eV. The SOCs and d-electrons in Mo play a very significant role in deciding its electronic and optical properties. Moreover, electronic elementary excitations are studied theoretically within the diagrammatic self-consistent field theory. Under random phase approximation, it shows that two branches of plasmon modes can be achieved via the conduction-band transitions due to the SOCs, which are different from the plasmons in a two-dimensional electron gas and graphene owing to the quasi-linear energy dispersion in single-layer MoS₂. Moreover, the strong optical absorption up to 10⁵ cm⁻¹ and two optical absorption edges I and II can be observed. This study is relevant to the applications of monolayer MoS₂ as an advanced photoelectronic device.