Caption: A Hubble diagram of galaxies for the very local universe out to ∼ 25 Mpc.

Features:

1. An ideal Hubble diagram is a plot of recession velocity versus cosmological proper distance for astronomical objects that participate in the expansion of the universe.

   The data points should according to theory all lie on the line defined by the theoretical Hubble's law:

     v = H_0*r  , 

   where v is recession velocity conventionally measured in kilometers per second (km/s), r is (cosmological) proper distance conventionally measured in megaparsecs (Mpc), H_0 is the Hubble constant which has modern fiducial values of 70 (km/s)/Mpc.
   Observationally determined values of H_0 vary by about 10 % from 70 (km/s)/Mpc nowadays (see Wikipedia: Hubble's law: Observed values).

   Note recession velocity v is NOT an ordinary velocity, but is the rate of growth of space between objects separated by proper distance r.

   Proper distance is true phyiscal distance which in principle can be measured at one instant in time with rulers.

   Hubble constant H_0 is the rate of universal expansion per unit proper distance. It is only a constant in space, NOT in time.

   When we speak of Hubble's law for general cosmic time, we call the coefficient in Hubble's law the Hubble parameter to indicate that it changes with cosmic time.

2. In fact, neither recession velocity nor proper distance are direct observable in general.

   In general, they must be obtained from a cosmological model fitted to observations.

   At present, the favored model is the Λ-CDM model.

3. An observational Hubble diagram as shown in the figure is what is usually meant by the expression Hubble diagram since that is a diagram that can be used to measure the Hubble constant and other parameters of cosmological models.

   The plot in figure is an observational Hubble diagram.

4. On an (observational) Hubble diagram, one plots redshift velocity versus a directly observable distance quantity, almost always luminosity distance---a distance quantity measured assuming a static observable universe.

   The given Hubble diagram calls redshift velocity "Redshift" for some reason.

5. Redshift velocity is given by v_red = z*c, where z is redshift and c is vacuum light speed.

   Redshift is a direct observable since z is.

6. Redshift is given by

     z = (λ_observed-λ_rest_frame)/λ_rest_frame  , 

   where λ_observed is the observed wavelength and λ_rest_frame is the rest-frame wavelength which is known for spectral lines.

   The redshift is due to two effects:

   1. cosmological redshift which is due to the growth of space stretching wavelength as a light signal propagates from source to observer.

   2. the Doppler effect (AKA Doppler shift) due to the peculiar velocities of source and observer.

   The peculiar velocities are usually ≤ ∼ 1000 km/s (corresponding to z ≤ ∼ 0.003), and so are always increasingly negligible as redshift velocity grows beyond say 3000 km/s (corresponding to z ≥ ∼ 0.01).

   The peculiar velocities may be negligible at even smaller redshift velocities depending on which astronomical objects you are observing.

7. The 1st-order in small cosmological redshift Hubble's law is

     v_red = H_0*r_observable  
   
     which if one knows what one means can be just written
   
     v = H_0*r   .  

   This formula in standard cosmological models (Friedmann-Lemaitre models which include the Λ-CDM model) can be used to determine H_0 with a relative error of order z for the cosmological redshift.

   So if you want to know Hubble constant H_0 to 1 % accuracy, you have to use objects at z ≤ 0.01.

   Unfortunately, peculiar velocities can cause errors of order up to ∼ 30 % for z ≤ 0.01 as we know from the discussion above.

   Fortunately, the peculiar velocities can be corrected for to some degree, and so nowadays fairly accurate values for the local Hubble constant are possible.

   One also has account for errors in measuring the distance values.

8. Note the expression "local Hubble constant" in the statement above.

   The 1st-order in small cosmological redshift Hubble's law is exactly accurate for exactly homogeneous Friedmann-Lemaitre models.

   But the real observable universe is NOT exactly homogeneous. It is believed to be so average over large enough scales which may have to ∼ 1 Gpc since the largest structures are of that size scale (see Wikipedia: Large quasar group).

   Determinations of the Hubble constant from Hubble diagrams may give us accurate local Hubble constant. but NOT the true global average Hubble constant.

   Fortunately, there are ways of measuring the global average Hubble constant.

   We won't go into them here.

9. A present it is thought that both the local and global average Hubble constant are 70 (km/s)/Mpc to within ∼ 10 %.

   Alas, determining them to higher accuracy is a very challenging problem.

   But people are working on this problem

   Still getting to consensus values of 1 % accuracy may take years, decades.