Waves are a way that energy is transferred from one place to another. There are two main types of wave, Transverse and Longitudinal. In both types of wave, energy is transferred in one direction, even though the particles just vibrate (either perpendicular or parallel to the direction of energy transfer).
Transverse
The particles in these waves vibrate perpendicular to the direction of the wave. That means that if the particles move vertically up and down, then the energy transfer is horizontal.
Examples of transverse waves include: a rope being waved up and down, water waves, plucked guitar strings, and electromagnetic (light) waves.
Longitudinal
The particles in these waves vibrate parallel to the direction of the wave. That means that if the particles move forward and back horizontally, the energy transfer is also horizontal.
Instead of particles moving up and down to make a wave, the particles move closer (compression) and further apart (rarefaction).
Examples of longitudinal waves include a slinky being pushed forward and sound waves.