Helioseismology is the study of the propagation of pressure waves in the Sun. Unlike seismic waves on earth, solar waves have practically no shear component (s-waves). Solar pressure waves are generated by the turbulence in the convection zone, near the surface of the sun, and certain frequencies are amplified by constructive interference. In other words, the turbulence "rings" the sun like a bell. The acoustic waves are transmitted to the outer photosphere of the sun, which is where the light emitted by the sun is generated. The acoustic oscillations are detectable on almost any time series of solar images, but are best observed by measuring the doppler shift of photospheric absorption lines. 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.
Helioseismology 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. The science can be compared to asteroseismology, which considers the propagation of sound waves in stars.
Solar oscillations are essentially divided up into three categories, based on the restoring force that drives them: acoustic, gravity, and surface-gravity wave modes.
The age of the sun can be inferred with helioseismic studies. This method provides verification of the age of the solar system gathered from the radiometric dating of meteorites.
Helioseismic observations revealed the inner uniformly-rotating zone and the differentially-rotating envelope of the Sun, roughly corresponding to the radiation and convection zones, respectively. The transition layer is called the tachocline.