Spectroscopy is the field of study that measures and interprets electromagnetic spectra. In narrower contexts, spectroscopy is the precise study of color as generalized from visible light to all bands of the electromagnetic spectrum.
An example of spectroscopy: a prism analyses white light by dispersing it into its component colors.
A huge diffraction grating at the heart of the ultra-precise ESPRESSO spectrograph.
UVES is a high-resolution spectrograph on the Very Large Telescope.
Atomic spectra comparison table, from "Spektroskopische Methoden der analytischen Chemie" (1922).
The electromagnetic spectrum is the full range of electromagnetic radiation, organized by frequency or wavelength. The spectrum is divided into separate bands, with different names for the electromagnetic waves within each band. From low to high frequency these are: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. The electromagnetic waves in each of these bands have different characteristics, such as how they are produced, how they interact with matter, and their practical applications.
A visualization of the electromagnetic spectrum.