The Lorentz force is the force of a magnetic field on a charged particle passing through. When a particle of charge q moves with velocity v in the presence of electric field E and magnetic field B, the force F is:
F = q (E + v × B)
This is derivable from Maxwell's equations but is not immediately evident.
The Lorentz force is an element of magnetohydrodynamics and is used in astrophysical models for situations including significant magnetism, such as solar and solar system phenomena. When modeling more distant events, where measurement of the magnetism is less clear-cut, magnetism may still be included as a means to produce a workable model, and inferred by the model's success.
Lorentz drag is the act of the Lorentz force causing material (e.g., plasma) to be moved, i.e., dragged and can be a factor in ionized atmospheres and a possible factor in Jupiter's. I believe the term MHD drag is used to mean the Lorentz drag, or possibly implies the sum of all the drags including the Lorentz drag, and other fluid-dynamic drags.