Solar oscillations are due to two restoring forces: first, the pressure gradient (e.g acoustic waves) and the other the Archimede force (e.g gravity waves). Both are created by turbulent motions. For acoustic modes (p-modes or 5min oscillations, frequencies around 3 mHz) it s in the upper layers of the Sun, and only about one hundred penetrate inside the nuclear core. With the recently launched spatial shuttle Soho, helioseismologists observe more than 20000 modes (over 1 000 000 predict with the theory), and can constrain the internal structure of the Sun. For gravity modes (g-modes, frequencies around 0.1 mHz and below) we suppose that there are created by the turbulent motions at the base of the convection zone (i.e 0.713 solar radius) as well as inside, but we are still looking for them because in the convection zone they are evanescent! If we discover them, we could constrain the nuclear core with more accuracy than with p-modes, and be able to evaluate the central temperature and the neutrinos fluxes as well as the Sun's variability and its impact on the Earth.
Not only solar type stars could oscillate, but most of them! Some stars have so big pulsations that their luminosity and surface radius vary a lot as for RR Lyrae or Cepheids stars. We call Asteroseismology the study of stars oscillations.
See for example this web site.