A longitudinal wave is a wave whose oscillation is parallel to the direction of propagation. Mathematically, the axis of oscillation is the parallel to the wavevector. The most typical example is acoustic waves, which are repeated compression and decompression of a solid material (usually air or water). When talking about mechanical waves specifically, such as acoustic waves, they are also called compression waves.
They are one of two categories of wave motion, the other being transverse waves.