A transverse electromagnetic (TEM) wave is an electromagnetic wave where both the electric (E) field and the magnetic (H) field vectors point perpendicular (transverse) to the direction of wave travel.
[2] Fig. 1 A TEM wave in motion
Neither field has a component along the propagation axis (z-axis), but this doesn't means the TEM itself doesn't have a direction to go to.
transverse: perpendicular
propagation axis: straight line/spatial path along which a wave/light beam/signal travels through space
Rope analogy By shake hand up and down a rope tied to the wall, the wave travels forward to the wall (propagation axis). But the rope itself only moves up and down, (the vector or peak of the motion is) perpendicular to the travel direction. The wave has a direction it is going—it just doesn't push the rope back and forth along that line.
A pure TEM wave's forward direction (z-axis) has no electric or magnetic field component (Ez = 0, Hz = 0). All force vectors point strictly sideways of each ohter.
A non-TEM wave is an electromagnetic wave with an electric or magnetic field component pointing in the direction the wave is traveling. Unlike TEM waves, where both fields sit completely sideways to the motion path, non-TEM waves include longitudinal components.
A wave with a component along the propagation (z) axis falls in either one of the 2 types:
TE wave (transverse electric):
TM wave (transverse magnetic):