Homologous collapse refers to a collapse of material such as a cloud of gas or dust such that at each point in time, the whole expanse of the collapsing cloud is at a single density. This is an ideal collapse scenario, used to approximate portions of an actual collapse. Given this ideal, which adheres to the effects of gravity but ignores counter forces such as gas pressure, the uniform density of the material grows over the course of the collapse, eventually all the material reaching a point in space at the same time. Newton's laws would make this happen during the collapse of a spherical cloud initially of a constant density if it weren't for the gas's pressure and the increase of that pressure with the increasing density and temperature. In a cloud of low-density gas (such as much of the interstellar medium) and sufficiently optically thin that it radiates away energy pretty much as soon it is unleashed by the collapse, a homologous collapse approximates much of the first portion of the collapse, and can serve as a workable model for the collapse of fairly uniform bodies, e.g., during the formation of stars, and possibly in some core collapse supernovae.
Such an ideal collapse has an easy-to-model structure, described by a relatively straight-forward equation. At any point in time, the further from the center, the higher the acceleration, due to more matter within the same radius from the center, which is exactly enough to keep the density (ratios) the same and bring all the matter to a point. In a real world collapse, potential energy released by gravitational collapse heats the gas and the combination of heat and density raise the pressure, slowing the collapse.