What we have termed “dark matter” is generic for observed gravitational effects on all scales: galaxies, small and large galaxy groups, clusters and superclusters, CMB anisotropies over the full horizon, baryonic oscillations over large scales, and cosmic shear in the large-scale matter distribution. The correct explanation or nature of dark matter is not known, whether it implies unconventional particles or modifications to gravitational theory. but gravitational effects have convincingly proved its existence in some form.
The few per cent of the mass of the Universe found as baryonic matter in stars and dust clouds is well accounted for by nucleosynthesis. If there exist particles which were very slow at time teq when galaxy formation started, they could be candidates for cold dark matter. They must have become non-relativistic much earlier than the leptons, and then decoupled from the hot plasma.
Whenever laboratory searches discover a new particle, it must pass several tests in order to be considered a viable DM candidate: it must be neutral, compatible with constraints on self-interactions (essentially collisionless), consistent with Big Bang nucleosynthesis, and match the appropriate relic density. It must be consistent with direct DM searches and gamma-ray constraints, it must leave stellar evolution unchanged, and be compatible with other astrophysical bounds.
The total dynamical mass of an astronomical system is derivable from the velocity dispersions or the rotation velocities of its components via the use of the Virial Theorem or Kepler's law, respectively. A most important probe is strong gravitational lensing which measures the total mass, but also weak lensing, the oscillations in the Cosmic Microwave Background and in the ambient baryonic medium. Probes separating dark matter from total matter require in addition observations of visible light, infrared radiation, X-rays, the Sunyaev-Zel'dovich effect, and supernovae. Depending on the system under study there are many ways to combine these tools using empirical halo models, simulating stellar population models and galaxy formation models, comparing mass-to-light ratios and mass autocorrelation functions. The most remarkable systems are merging galaxy clusters which, by their motion, separate non-collisional dark matter from optically visible galaxies and hot, radiating gas.
Regardless of the nature of dark matter, all theories attempting to explain it share the burden to explain the gravitational effects described in here. Thus there remains much to be done.
I am grateful to Sylvain Fouquet, Carmen Rodríguez-Gonzálvez, and Will Dawson for clarifying comments and addenda.