Spectroscopy (or Spectrometry) is the study of radiation intensity as a function of Wavelength. It is the method for determining the chemical composition of astronomical bodies as well as Temperature and radial motion. Spectrography is virtually the same study but implies the use of a spectrographic image, displaying the spectrum across its width.
Instruments operating at or near Visible Light typically use prisms or Gratings that angle light according to wavelength (Dispersion). Dispersion with prisms is small so often multiple prisms are used in tandem, e.g., a Triple Prism Spectrograph.
Photometry is like an extremely-low-Resolution spectroscopy, studying stars and astronomical bodies based upon just a few Bands. Its advantage is that it requires much less Electromagnetic Radiation, thus can be used for more distant objects, and is also multi-object by default.
Some instrument terms/classes:
Intensity at each wavelength is typically the item of interest but there are also cases when Polarization at each wavelength if the item of interest.
Chandra X-ray Observatory (CXO)
Discovery Channel Telescope (DCT)
Velocity Dispersion (σ)
Exoplanet Eclipse Light Curve
Gran Telescopio Canarias (GTC)
Imaging Fourier Transform Spectroscopy (IFTS)
Integral Field Spectrograph
Nearby Supernova Factory (NSNF)
Rossiter-McLaughlin Effect (RM Effect)
Supernova Cosmology Project (SCP)
Stellar Parameter Determination
Stellar Temperature Determination
2dF Galaxy Redshift Survey (2dFGRS)
James Webb Space Telescope (JWST)
William Herschel Telescope (WHT)