Asteroseismology is the study of stellar oscillations and describes the behaviour of pulsating stars. We see most stars only as tiny bright spots in the night sky although, in reality, they are huge sizzling hot gaseous balls. The sun, for example, has a diameter of roughly 1.4 million kilometres. As such, it would take the fastest car at a maximum speed of 508,5 km/h about 114 days to pass it once. This is of course impossible not only because the sun's temperature is about 5000˚C at its surface and a whopping 15 Million˚C in its core.
It is not only the temperature, but also the forces that rise to stellar values. The gravity and pressure are main contributors to what controls the structure of stellar interia. These forces power the oscillations of stars, giving rise to the two main families of stellar pulsation: gravity- and pressure modes.
Stars spend most of their lifetimes in the status of hydrostatic equlibrium. That is, the gravitational pull on a spherical layer of the star is balanced by the force from the pressure gradient. Hydrostatic equilibrium plays a major role in thermal stability, one of the three fundamental equilibrium configuration of stars (Unno et al. 1989): Thermal stability, dynamical stability and vibrational stability. If the star is thermallly stable, excess heat will cause the star to expand and cool as a consequence to hydrostatic adjustment. However, if there is thermal instability, excess heat can lead to a runaway reaction. This is for example the case for the Helium Flash.
A dynamical stable system will always oscillate around its equilibrium position if a small force is applied. It is however the time dependence of this oscillations that decide wether a star is vibrationally stable or not. If the amplitude of the oscillations grows with time, the star is vibrationally overstable or unstable. The star will however never oscillate randomly. The governing forces, and especially the resulting physical equations, allow only specific pulsation modes to obtain high amplitudes, the eigenfunctions. This pulsation modes are spherical harmonics $ Y^m_l $ together with a time dependence $ exp(i\omega t)$.
The simplest case a radial pulsations, with $m=l=0$. Here, the star contracts and expands radially. It is important to state however that their exists a radial order, $n$. If this $n=1$ the contraction and expansion happens similarily for the whole star. This is similar to the case of a string that is fixed at both ends and swings up and down. Higher values of $n>1$ correspond to higher number of oscillation nodes. For $n= 2$ there is one oscillation node in the middle of the string: The part left goes up while the part on the right goes down. It is no different for a radial pulsating star: The center contracts while the surface expands and vice versa.
Real stars often pulsate in multiple pulsation modes. Their respective expansion and contraction area will hence add up and will look much more complicated. In addition, their periodicity (i.e. the frequency) is typically different. Even more: the spherical harmonics shown here are only true for non-rotating stars. In reality, most stars rotate which will lead to (at least some) oblateness and hence also change eigenfunctions of the oscillations.