A unified electrodynamic theory of magnetorotational and related instabilities in a rotating plasma is formulated. We consider a cylindrical plasma in a uniform longitudinal magnetic field with rotation sustained due to the radial gravitation force or/and radial plasma pressure gradient. Two marginal cases are analyzed: the simplest astrophysical one without the pressure gradient and the laboratory plasma without the gravitation force. The perturbations are described within the one-fluid magnetohydrodynamics (MHD) and the kinetics. For such equilibria and for both the approaches a universal local dispersion relation is derived in terms of the permittivity tensor. Four versions of this tensor are obtained: for the simplest MHD and kinetic astrophysical plasmas with constant pressure and for the general MHD and kinetic plasma models. In the simplest MHD astrophysical plasma model, the axisymmetric and nonaxisymmetric magnetorotational instabilities (MRIs) and convective instability are considered. It is shown that the nonaxisymmetric modes are less dangerous than the axisymmetric ones due to the strongly stabilizing magnetoacoustic effect and also because of the overstable effect. We found that the one-fluid model for nonaxisymmetric modes in long laboratory plasma predicts the enhanced effect of the pressure gradient, compared with the case of axisymmetric modes, as the squared length/width ratio of the system. The axisymmetric and nonaxisymmetric kinetic MRIs in a collisionless plasma with isotropic pressure are analyzed. It is shown that the latter instability looks like an overstability driven for the same condition as the axisymmetric one. The pressure anisotropy of the rotating plasma gives two hybrid instabilities: the rotational firehose and rotational mirror. For both the astrophysical and laboratory cases the pressure gradient effects in the kinetic model are considered and a family of axisymmetric and nonaxisymmetric kinetic pressure-gradient-driven instabilities is found.