The basic observation which a non-standard cosmology needs to deal with the observation of cosmic redshift. The standard cosmology says that the cosmic redshift is due to galaxies receding away from each other and that the recession of the galaxies is due to general expansion of space as predicted by general relativity.
A non-standard cosmology challenges one or both of these beliefs. For example, a non-standard cosmology may attribute the redshift to physical effects of photons or it may assert that the universe is expanding but that this expansion is not the result of space expansion (i.e. the Alfven universe, see Plasma cosmology).
Alternative models of cosmology that do not challenge these two assertions are not termed non-standard cosmologies, even if they are not widely accepted. For example, the modified Newtonian dynamics is not a non-standard cosmology even though it radically challenges the mainstream views on gravity.