What's the difference between longitudinal and transverse waves?

Firstly, all waves transfer energy without transferring matter (particles). 

Waves usually consist of particles vibrating in a particular direction, but not necessarily in the same direction as the way the energy is transferred. For example - a water wave. The water moves up and down, but the wave (or energy) might be moving to the right.

For transvere waves - the particles vibrate perperndicular to the direction of energy transfer (eg. water)

For longitudinal waves - the particles vibrate parallel to the direction of energy transfer (eg. air particles colliding causing a sound wave)