Vorticity plays a crucial role in fluid dynamics, so that it is quite important to understand how vorticity is generated. It is usual to describe the evolution of vorticity in terms of tilting and stretching of a vortex line. The same explanation is applied to absolute vorticity in rotating fluids. In oceanography and meteorology, however, relative vorticity is more straightforward than absolute vorticity. There seems to have been some confusion or ambiguity as regards the generation mechanism of relative vorticity in rotating fluids. This article is intended to present a comprehensive and comprehensible picture of the total processes of vorticity generation, both for non-rotating and for rotating fluids. That is, another kind of explanation of vorticity dynamics is proposed based on a concept like the torque on a rigid body. We define vorticity force in a local and instantaneous frame of reference, which is moving and rotating uniformly at the translation velocity and angular velocity of each fluid particle, respectively. In short, vorticity force is the Coriolis force in the local frame of reference transpating and rotating as above. It is possible to interpret that the twisting force or torque due to vorticity force changes the vorticity of the fluid particle in concern. This explanation is mathematically rigorous, general, and causal, so that we understand the mechanism of vorticity generation intuitively and visually. Although twisting by vorticity force is most important in fluid dynamics, there are other factors that change the vorticity of fluid particles such as external force or viscous force. As an example or application, we illustrate the balance of vertical vorticity in the subtropical ocean of the northern hemisphere, where β-effect and viscous force contribute to that balance. Finally we clarify the meaning of baroclinic twist and its relation with the thermal-wind balance; we see that baroclinic twist is not the twist of force, but the twist of acceleration. Also we argue the Boussinesq approximation rather in details, where buoyancy twist is shown to be an approximate form of baroclinic twist; it generates only the horizontal components of vorticity.