Helioseismology is the study of how pressure waves propagate in the Sun. These waves are measured by the doppler shift of visible solar material. Changes in the propagation of pressure waves through the Sun reveal inner structures and allows astrophysicists to develop extremely detailed profiles of the interior conditions of the Sun.
Helioseismology was able to rule out the possibility that the solar neutrino problem was due to incorrect models of the interior of the Sun. Features revealed by helioseismology include that the outer convective zone and the inner radiative zone rotate at different speeds to generate the main magnetic field of the Sun, and that the convective zone has "jet streams" of plasma thousands of kilometers below the surface. These jet streams form broad fronts at the equator, breaking into smaller cyclonic storms at high latitudes.
It can also be used to detect sunspots on the far side of the Sun from Earth.
Keep in mind that despite the name, Helioseismology is the study of solar pressure waves and not solar seismic activity - there is no such thing. The name is derived from the similar practice of studying terrestrial seismic waves to determine the composition of the Earth's interior.
See also