A spectral signature is a characteristic or set of characteristics of an emission's spectral energy distribution that reveal something specific about the material or conditions that produced the electromagnetic radiation. Typical would be characteristics suggesting the material and temperature such as a fit to the Planck function along with particular spectral lines. A spectral index over some regime might be a signature for some type of emission, e.g., synchrotron radiation.
There are spectral signatures associated with certain types of stars (spectral classes), and additional characteristics, such as metallicity. Similar can be said of galaxies, quasars, clouds, planets, etc.