A transverse wave is a wave whose oscillation is perpendicular to the direction of propagation. Mathematically, the axis of oscillation is the orthogonal to the wavevector. The most typical example is electromagnetic waves, which are coupled oscillating electric and magnetic fields. In three dimensions, there are infinite possible perpendicular axes to the propagation, so there is a certain freedom in what direction the transverse wave can oscillate in. It may oscillate chaotically in no particular axis or it may have a preferred axis; in the latter case, the axis is called the polarization vector.
They are one of two categories of wave motion, the other being longitudinal waves.