Waves transfer energy without transferring matter. Waves can also be used to transfer information, e.g. mobile phones or radios.
Mechanical waves are vibrations that travel through a medium (a substance). Examples include sound waves, water waves, waves on springs and ropes, and seismic waves produced by earthquakes.
Electromagnetic waves can travel through vacuums at the same speed of 300,000km per second. They do not require a medium. Examples include light waves, radio waves and microwaves.
The oscillations of a transverse wave are perpendicular to the direction in which the waves transfer energy. All electromagnetic waves are transverse waves.
The oscillations of a longitudinal wave are parallel to the direction in which the waves transfer energy.
Mechanical waves can be transverse or longitudinal.
Sound waves travelling through air are longitudinal waves. When an object vibrates in air, it makes the air around it vibrate as it pushes and pulls on the air. The oscillations (compressions and rarefactions) that travel through the air are sound waves. The oscillations are along the direction in which the wave travels making it longitudinal.