Comoving units are units of measurement that divide out the Hubble constant (in the form of the Hubble parameter). Comoving distance is basic, and from it, any measure based on distances, areas, or volumes can be expressed in comoving units.
Two distant objects observed in the past, e.g., galaxy clusters, can be further apart today, purely due to Hubble expansion. Comoving distance expresses the distance between them as a distance times the Hubble parameter, to ignore or cancel that effect. It is used, when appropriate, to compare phenomena at different cosmological times, and also to allow for future refinements of the Hubble constant that would affect determinations made from distant observations.
Comoving coordinates are coordinates using comoving distance and comoving space density is density (e.g., of mass per unit volume) based on the same. Comoving distance is also called coordinate distance.
A similar kind of coordinates can be used in describing black holes, with coordinates moving inward. Coordinate speed is the rate at which the coordinates shift.