Vasudevan Mukunth in The Wire:
A topological phase of matter is one whose topology and energy are related. For example, physicists have known that at a lower temperature, the surface of a single-atom-thick layer of superfluid helium develops vortices in pairs that move around each other according to how they are both rotating. At a slightly higher temperature, the vortices become unpaired – but stay put instead of moving around. This is a topological phase transition: the topology of the substance changes according to the temperature.
This exact example – of vortices in liquid helium – is called the Kosterlitz-Thouless (KT) transition, for its discoverers David Thouless and John M. Kosterlitz. There are many other examples of topological phase transitions, all utilising the quirky things that quantum mechanics makes possible in strange, sometimes useful, ways. For example, physicists use topological concepts to understand electrical conductors better (especially insulators and superconductors), as well as apply it to the study of the smallest packets of energy as well as to discover the shape of the universe. In engineering, topological phases are used to find particles that, when they bump into others of their own kind, vanish in a puff of energy; build more efficient hard-drives; and make better robots.
This breadth of applications, as the British-American physicist F. Duncan Haldane has remarked, is thanks to quantum mechanics’s willingness, and classical physics’s reluctance, to be bizarre.
More here.