What are the conditions for cooling and collapse to be important?

Consider a collapsing gas cloud.

If the cooling time is shorter than the collapse time, as the cloud collapses it will efficiently radiate away the gravitational energy, cooling and continuing to collapse.
If the cooling time is longer than the collapse time, the cloud cannot radiate away the gravitational energy, so it will heat up and thermal pressure will stop its further collapse.

So for collapse, cooling, star formation, galaxy formation, etc, we want t_cooling<t_collapse

Consider a density-temperature plot for the collapsing object:

Based on the cooling curves, we can plot lines where t_cooling=t_collapse; for densities above this line, cooling is efficient.
For a given density and temperature, we can work out the virial mass of an object, and plot lines of constant mass.
For a given redshift, we can work out the typical density of a virialized object.

(from Galaxy Formation and Evolution; Mo, van den Bosch, and White)

Depending on metallicity, systems with masses below 10¹¹ - 10¹² M_sun can collapse and cool; higher masses cannot. Hot X-ray emitting gas in galaxy clusters today cannot cool very efficiently and remains hot.

(Note also that at low temps atomic cooling is inefficient. Need molecular cooling, and forming molecules is hard at low metallicity.)

Remember, though: This process describes cooling and collapse of a single gas cloud ("Monolithic Collapse") or of hot gas slowly accreting into a halo ("Hot Flow Accretion"). Galaxies can also grow (and mostly do) through mergers of smaller galaxies ("Hierarchical Collapse") or accretion of already-cooled gas ("Cold Flow Accretion"). So we need to turn to more complex considerations of galaxy formation.