The concept of Dark Matter is used to explain the apparent additional Gravity of a Galaxy, Galaxy Cluster, or other large structures beyond that explained by the visible and inferred stars, Interstellar Media and other normal matter. The amount of gravity is inferred by the velocity of the orbiting stars, determined by examining Doppler Shifts of the starlight.
The front-line theory is that Dark Matter must consist of otherwise-undiscovered weakly interacting massive particles (WIMPs) "weakly interacting" meaning, for example, that photons generally don't interact with them, thus we can't see it. Surveys have not borne out the competing MACHO theory, that there are sufficient "free floating" planet-sized objects to account for the apparent extra mass.
Dark Matter theories are also classified as either Cold Dark Matter, Warm Dark Matter, or Hot Dark Matter.
Without Dark Matter, some other explanation must exist for the apparent matter in galaxies. Theories have been proposed that the known laws of gravity (General Relativity) are only an approximation that breaks down in the regime of galaxies, e.g., Modified Newtonian Dynamics.
Cold Dark Matter (CDM)
Cosmic Evolution Survey (COSMOS)
Critical Density (ρc)
Conditional Stellar Mass Function (CSMF)
Dark Matter Halo
Halo Abundance Matching (HAM)
Lambda-CDM model (ΛCDM)
Modified Newtonian Dynamics (MOND)