The galaxy power spectrum is a power spectrum representing the density of galaxies across space.
A power spectrum is a representation of the magnitude of various frequency components. For galaxy power spectrums, the magnitude is the probability of finding a galaxy, or the number of galaxies, and the "frequency component" is distances from the galaxy, the spectrum showing for each possible distance from a galaxy, how likely it is that another galaxy is at that distance, for all the galaxies being analyzed. E.g., for 100Mpc, it shows as a probability density, how likely it is to find two galaxies that distance apart, the spectrum being graphed just on a range of such distances. Real galaxy power spectra can be produced from galaxy survey data such as 2dF Galaxy Redshift Survey, and theories (e.g., baryon acoustic oscillations) can be developed to match or can be tested according to whether they match.
The "ultimate" is a galaxy power spectrum covering three dimensions, using redshift to determine how far the galaxies are, which demands surveys that find galaxies to far distances red with some means to accommodate bias, along with redshift determinations. Simpler spectra based on two dimensions in space or one dimension are also used, based on slices of the volume, or based merely on direction (celestial coordinates), or on direction, at a redshift slice (part of a sphere of space roughly equidistant from us).
The matter power spectrum is a similar concept, but based on mass rather than the presence of a galaxy. Theories and simulations can provide them in theory. One step toward finding the "real" spectrum is to theorize and apply a correction factor relating it to a galaxy power spectrum.