Sound waves in air are waves of compression and decompression (called rarefaction) generated by a vibrating surface, which alternately pushes and pulls at the neighboring air, the waves then travel outwards. We hear sound because the rapid succession of compressive pulses causes the eardrum to vibrate at the same frequency, which is between 15 cycles per second and 20,000 cycles per second for young ears.
A main point of this animation is to make clear that although the movement of waves to the right is clear, there is no net transport of the air: toggle the red line to see that any "slice" of the air just vibrates about its original position, no air moves to the right except momentarily, then it moves back. Of course, there is movement of energy to the right: it takes energy to make your eardrum vibrate.
In an actual sound wave, the density variation is a lot smaller than that shown here.
You can find my lecture on sound waves here.
Code by Nicholas Anderson.