Longitudinal magnetism and transverse magnetism are components of the net magnetism vector.
Longitudinal magnetization is the component of the net magnetization vector parallel to the magnetic field (z-axis). This is due to a difference in the number of spins in parallel (low energy) and anti-parallel (high energy) state – i.e. before a radiofrequency pulse.
Transverse magnetization is the component of the net magnetization vector perpendicular to the magnetic field (x-y plane). This is due to the spins of individual protons getting more or less into phase (coherence) – i.e. after a radiofrequency pulse.