What is the difference between transverse waves and longitudinal waves

In a transverse wave, the vibration of the particles (or the direction of the oscillation in the case of electromagnetic waves) is perpendicular to the direction of propagation. This means, if the wave moves in the x-direction, the oscillations may be in the y-direction, for example, the plane waves are transverse waves.

Longitudinal waves, as the name says, are the waves where the oscillation is parallel to the direction of movement. In the case of mechanical waves, this type is called "compressional waves", an example of this can be the pressure waves or the sound waves. For the electromagnetic waves this case is not really common, but Maxwell's equations do lead to the appearance of longitudinal waves under some circumstances, for example, in plasma waves or guided waves.