The Coriolis force is what seems like a force to an object moving within a rotating frame of reference. As the object is curving according to the frame of reference, in that frame of reference, it seems to be forced to one side. It is a factor in rotating fluids, e.g., atmospheres, stars, disks, clouds in galaxies, etc. The apparent force is to the left if the rotation is clockwise, and to the right if it is counterclockwise. The apparent deflection of the object is called the Coriolis effect.
A beta plane is a planar approximation of a surface affected by the Coriolis force. It is used to model phenomena without all the complications of a spherical surface, and is presumably most useful for phenomena with a span much smaller than the size of the spherical surface