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The Vlasov-Poisson equation is a simplified equation for plasma, somewhat like the Maxwell-Boltzmann equation is for ideal gases, but not presuming equilibrium: it describes an evolution of an ideal plasma's distribution. It is a simplified version of the Vlasov-Maxwell equation. These equations model the distribution of gases of charged particles in which electromagnetic forces are significant even at some distance. Essentially, they are the Boltzmann transport equation (BTE) with no collision term (the collisionless Boltzmann equation, which assumes the elastic collisions corresponding to those of neutral gas are insignificant) and with its force term describing the electromagnetic fields of that instant due to the moving charged particles. The Vlasov-Poisson equation's simplification is to ignore relativity as well as any effect of magnetic fields. Either of these equations must be used simultaneously for each type of charged particle. In the case of the Vlasov-Poisson equation, they are used in conjunction with a Poisson's-equation description of the electric field, the whole equation set termed a Vlasov-Poisson system. All these equations are likely to require numerical methods for any application and further simplifications can be useful.
Usage of the simpler phrase Vlasov equation varies: it has been used for each of the above "Vlasov" equations, and sometimes even for the collisionless Boltzmann equation. The term Vlasov simulation is a simulation, e.g., of plasma, making use of one of these Vlasov equations, generally codes taking into account Coulomb force, and often ignoring collisions, which can be insignificant in the thin plasma common in astrophysical phenomena. The term Vlasov solver is also used for such codes, suggesting the calculation is merely solving equations describing the process.