To add a little to Ryan's answer: 

This topic is maybe not exactly a part of algebraic topology. It's more like an area of application of algebraic to certain important special classes of spaces. 

When $A$ is a subcomplex of $B$ (both of them finite) and $B$ collapses to $A$, then the inclusion map $A\to B$ is said to be a simple homotopy equivalence, as is any left inverse of such an inclusion, and more generally any map between finite complexes that is homotopic to a composition of such things. A homotopy equivalence $A\to B$ between finite complexes determines an element of the Whitehead group $Wh(G)$ of the fundamental group $G=\pi_1(A)$ (a certain abelian group that depends functorially on $G$ -- the quick definition is take the direct limit of $GL_n(Z[G])$ as $n$ goes to infinity, abelianize, and kill the the invertible $1\times 1$ matrices $g\in G$ and $-1$). It is simple if and only if this element is zero. (Sometimes the latter is taken as definition of simple.) The group $Wh(1)$ is trivial, so every homotopy equivalence between simply-connected finite complexes is simple. The house with two rooms shows that the inclusion of a subcomplex can be simple even if there is no collapse; there is a larger complex collapsing both to the house and to the point.

The question of whether a homotopy equivalence is simple is unchanged by subdivision of a complex, so sometimes you can prove that a given homotopy equivalence is not homotopic to any simplicial or cellular isomorphism by proving that it has nontrivial Whitehead torsion. If you can enumerate all the homotopy equivalences from $A$ to $B$ and find that all of them have nontrivial torsion, then the spaces are really different. Eventually the toplogical invariance of Whitehead torsion was proved, making for stronger statements.

$Wh(G)$ is trivial for lots of (potentially for all) torsion-free groups $G$, but usually nontrivial for finite $G$.