Let $G_1 \subset G_2 \subset \dots$ be a sequence of groups and inclusion homomorphisms. It satisfies homological stability if for every $r \in \mathbb{N}_0$ there is an $n(r)$, such that for all $n > n(r)$ the homomorphism $$ H_r(G_n) \to H_r(G_{n+1}) $$ induced by the inclusion is an isomorphism. Examples of groups that have homological stability are the symmetric groups $S_n$, the braid groups $B_n$ and the automorphism group of the free groups $Aut(F_n)$.
Harer proved that the mapping class groups $\Gamma_{g,b}$ of surfaces $\Sigma_{g,b}$ of genus $g$ with $b$ boundary components given by $$ \Gamma_{g,b} = \pi_0(Diff^+(\Sigma_{g,b}, \partial \Sigma_{g,b})) $$ satisfy homological stability in the sense that the homomorphisms of the homology groups obtained from gluing on various surfaces along the boundary components eventually become isomorphisms. For example, gluing on a pair of pants for $b \geq 1$ gives a homomorphism $$ \Gamma_{g,b} \to \Gamma_{g,b+1} $$ which induces an isomorphism in homology $H_r(\Gamma_{g,b}) \to H_r(\Gamma_{g,b+1})$ in the range $r > 1$, $g \geq 3r-2$.
These results were later generalised by Madsen, Weiss, Galatius, Randal-Williams, Hatcher, Vogtmann, Wahl and many others. It has influenced the way to think about cobordism categories and has revealed a lot about the structure of moduli spaces of surfaces (possibly with tangential structures).
This answer is much too short to give a huge and fruitful subject the attention it deserves. But maybe some of the people working in this area (some of which are active on mathoverflow) can expand on it.
