Waves can be classified by the direction in which the local disturbance moves compared with the direction in which the wave propagates.

Transverse waves

In a transverse wave, the disturbance is perpendicular to the propagation direction.

A wave travelling horizontally along a stretched string can make the string move vertically. The wave moves along the string while each small element moves across that direction.

Transverse waves can have different orientations of their disturbance, which makes polarization possible.

Longitudinal waves

In a longitudinal wave, the disturbance is parallel to the propagation direction.

Sound in a fluid is the main example: regions of compression and rarefaction travel through the medium while the fluid particles oscillate mainly back and forth along the same direction.

The medium does not travel with the wave

In either case, the local particles usually oscillate around equilibrium rather than moving continuously with the wave over long distances.

The distinction concerns the geometry of the disturbance, not whether the wave is mechanical or electromagnetic.