Sound waves are mechanical waves, which means they transfer energy through a medium, such as air, water, or some metal. The way sound waves do this is through variations in pressure through said medium.

In fluid media (such as air or water), this phenomenon is manifested in the form of "longitudinal waves." This means the oscillations of the waves are in the same direction they're travelling. To picture it, you can think of the tide coming in and leaving the beach.

In solid media, however, sound waves can travel as both longitudinal and transverse waves. Transverse waves are those in which oscillations occur at a 90 degree angle from the direction of travel. They typically look like some combination of sines and/or cosines. It turns out there are other "shapes" of waves in which sound can travel, but these are more complicated and are usually formed by some combination of longitudinal and transverse waves.

In both cases, sound waves periodically push on the molecules that comprise the media (e.g. air molecules, or the molecules that make up a violin string). The way we end up hearing sounds is because our ears have tiny, extremely sensitive hairs that detect changes in air pressure. The hair cells move mechanically, and "talk" through electrochemical and neurotransmitter signals to a series of neurons that eventually relay the sound information to the part of your brain that interprets sounds!

Sound is a vibration and the waves are longitudinal waves. The waves are created by the vibration of an object which causes air to vibrate. Air vibration causes your eardrum to vibrate and finally your brain interprets that sound. Sound waves travel through air just how you would think regular water waves travel in water.