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Comoving units are units of measurement that divide out the scale factor; in other words, the units themselves are growing along with Hubble expansion.
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 scale factor at that time, to factor out 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: e.g., comoving volume is the comoving distance cubed, and comoving space density is the mass per comoving volume. Comoving distance is also called coordinate distance.
Another 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.