WAVES
In waves, energy propagates without the propagation of mass. In other words, the particles that constitute the wave oscillate around equilibrium positions so that they don't have a net translation.
In the case of electromagnetic waves, there is no need of mass. This waves propagate even in the vacuum.
Longitudinal waves
The particles oscillate in the same direction in which the wave propagates. A typical example are sound waves. The animation below illustrates how sound waves propagate on air.
Repair how the red bars oscillate around equilibrium positions, so that there is no net movement. They represent air molecules.
Transversal waves
The particles oscillate in the direction perpendicular to that which the wave propagates. A typical example is a wave in a rope:
