While general relativity allows one to formulate the laws of physics using arbitrary coordinates, some coordinate choices are natural choices with which it is easier to work. Comoving coordinates are an example of such a natural coordinate choice. They assign constant spatial coordinate values to observers who perceive the universe as isotropic. Such observers are called "comoving" observers because they move along with the Hubble flow.
A comoving observer is the only observer that will perceive the universe, including the cosmic microwave background radiation, to be isotropic. Non-comoving observers will see regions of the sky systematically blue-shifted or red-shifted. Thus isotropy, particularly isotropy of the cosmic microwave background radiation, defines a special local frame of reference called the comoving frame. The velocity of an observer relative to the local comoving frame is called the peculiar velocity of the observer. Most large lumps of matter, such as galaxies, are nearly comoving, i.e., their peculiar velocities are low.
The comoving time coordinate is the elapsed time since the Big Bang according to a clock of a comoving observer and is a measure of cosmological time. The comoving spatial coordinates tell us where an event occurs while cosmological time tells us when an event occurs. Together, they form a complete coordinate system, giving us both the location and time of an event.
Space in comoving coordinates is (on the average) static, as most bodies are comoving, and comoving bodies have static, unchanging comoving coordinates.
The expanding Universe has an increasing scale factor which explains how constant comoving coordinates are reconciled with distances that increase with time.
Comoving distance is the distance between two points measured along a path defined at the present cosmological time. For objects moving with the Hubble flow, it is deemed to remain constant in time. The comoving distance from an observer to a distant object (e.g. galaxy) can be computed by the following formula:
If one divides a comoving distance by the present cosmological time (the age of the universe) and calls this a "velocity", then the resulting "velocities" of "galaxies" near the particle horizon or further than the horizon can be above the speed of light. This apparent superluminal expansion is not in conflict with special or general relativity, and is a consequence of the particular definitions used in cosmology. Note that the cosmological definitions used to define the velocities of distant objects are coordinate dependent - there is no general coordinate independent definition of velocity between distant objects in general relativity (Baez and Bunn, 2006) The issue of how to best describe and popularize the apparent superluminal expansion of the universe has caused a minor amount of controversy. One viewpoint is presented in (Davis and Lineweaver, 2003)
Within small distances and short trips, the expansion of the universe during the trip can be ignored. This is because the travel time between any two points for a non-relativistic moving particle will just be the proper distance (i.e. the comoving distance measured using the scale factor of the universe at the time of the trip rather than the scale factor "now") between those points divided by the velocity of the particle. If the particle is moving at a relativistic velocity, the usual relativistic corrections for time dilation must be made.