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They use the Red Shift to determine that distance.

The red shift is just a first-order way to estimate the distance. If the Hubble Parameter (which used to be called the Hubble Constant when people thought it was constant) was a simple constant, that's a straightforward calculation. When it became more complex it becomes more of a table lookup, but as a first order it's still fairly valid. By that, it means that for items without some better way to estimate their distance, it's a good start.

The best way to estimate the distance to some deep space object is by measuring the apparent brightness of something with a known absolute or intrinsic brightness. The article mentions Cepheid variables which are a very good way to do that, and form the basis for Edwin Hubble's first recognition that the universe was much larger than the Milky Way, and eventually to identify the Hubble Constant. However, in detail it is known to have limitations because some galaxies are influenced by galactic clusters and so their relative motion is not solely due to the expansion of the universe. For example, Andromeda galaxy is approaching us, not receding.

Type 1A supernovae are the most distant distinctly observable phenomena for which we believe we know the intrinsic brightness and enough about the spectrum that we can determine both their absolute distance (from apparent versus intrinsic brightness) and relative velocity (from red shift of the spectrum). They are the main reason we now think the Hubble "Constant" is not really constant, but drops off with distance (or, conversely, increases as you get closer in distance and time). That accelerating expansion is not explained by gravity - in fact, it contradicts the expected effect of gravity. Since the velocity (and hence energy) of observed objects seems to be increasing, they call the mechanism we can't otherwise detect "dark energy."

Thanks for taking the time to type up that reply. Happily I knew quite a bit of the basics of distance measurement.

They are the main reason we now think the Hubble "Constant" is not really constant, but drops off with distance (or, conversely, increases as you get closer in distance and time). That accelerating expansion is not explained by gravity - in fact, it contradicts the expected effect of gravity. Since the velocity (and hence energy) of observed objects seems to be increasing, they call the mechanism we can't otherwise detect "dark energy."

This makes me scratch my head. If the Hubble constant drops off then that implies a slower expansion at the outer edge. Would they expect gravitational slowdown to be equal at all distances? Therefore over a billion years the observed Hubble Constant would reduce evenly. I guess only if they expect even distribution of matter.

I remember my old astronomy books suggesting that galaxies were evenly distributed. How surprised they were to find clusters, filaments, voids and strange motions among them.