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It regroups the notion of Berry's phase, Aharonov-Bohm and Aharonov-Casher type of phase among others. It is mathematically described by the notion of the (an-)holonomy group, associated to the vast mathematical notion of connection on a finer-bundle.
The use of topological phase arises in almost all the areas of modern physics, ranging from continuous mechanics (elastic problems, ...) eventually described by statistical physics methods (fluid mechanics, polymer physics, ...) to high-energy physics passing through the quantum-side of condensed matter (defects in ordered media, fluxoid quantisation, two-dimensional problems, quantisation of Hall conductance ...) and mesoscopic problems (adiabatic pumping, proximity effect, ...)
It should be not confounded with the notion of topological order (i.e. the word phase has to be understood as the phase of the wave-function, not as any kind of phase of matter like in the term phase transition which a topological phase actually doesn't have). The appearance of a topological phase is a (necessary, not sufficient) prerequisite for the appearance of topological order.