Waves don't have mass because they aren't matter. A wave is a disturbance that travels through a medium by displacing the particles of the medium without transporting them. Electromagnetic waves such as light can travel through empty space. Waves are a form of energy.
There are two types of wave propagation - transverse waves and longitudinal waves. Transverse waves displace matter in a direction that's perpendicular to the direction it travels. Think about a wave in water moving the water up and down as the wave propagates horizontally. After the wave passes the medium is still there. Light movement is also transverse although it doesn't displace matter.
Longitudinal waves displace the particles of the medium in the direction of propagation. Imagine that you and a friend are each holding the end of a stretched slinky (or large metal spring) and you quickly push it straight forward and pull it back. You will create a wave that travels the length of the slinky, temporarily compressing each area it passes through. Sound waves are an example of longitudinal waves.