What is non geodesic motion?
Just to elaborate quickly on what HallsofIvy said: in general relativity, the world line of a (nonspinning) test particle is a geodesic, i.e. a curve with zero path curvature. More generally, the path curvature at any event on some world line is just the magnitude of acceleration experienced by the corresponding particle. For example, a charged particle in an "electrovacuum solution" of the Einstein field equation (EFE) will experience a nonzero Lorentz force, whose magnitude (at each event on the world line of the particle) agrees with the path curvature.
Why spinning particles don't have a geodesic path?