What does a distant galaxy having a greater red shift indicate about the universe?

Answer 1

A red-shift caused by the Doppler-effect indicates that the object is rapidly moving away from the observer.

Note that at extreme distances, red-shift is not caused by the Doppler-effect, but primarily by the expansion of space between the object and the observer.

Answer 2

There is no specific "red shift theory"; redshift is an observational fact, meaning that light arrives with the energy levels shifted towards less energy, compared to the way it was emitted. This can be observed in the light's spectrum.
Redshift means that the light has somehow lost energy before it reached us; for example, through the Doppler effect, through the expansion of space, or because it was emitted from a "gravitational well".

Answer 3

This is usually attributed to the Doppler effect (there doesn't really seem to be any other reasonable explanation); in which case, the conclusion is that the Universe is expanding.

Answer 4

Speed an object is moving away from the observer.

Answer 5

The "Doppler Theory".