Topological order is a new kind of order in zero-temperature phase of quantum spins, bonsons, and/or electrons. The new order corresponds to pattern of long-range quantum entanglement. Topological order is beyond the Landau symmetry-breaking description. It cannot be described by local order parameters and long range correlations. However, topological orders can be described/defined by a new set of quantum numbers, such as ground state degeneracy, non-Abelian geometric phases of degenerate ground states, quasiparticle fractional statistics, edge states, topological entanglement entropy, etc.
Fractional quantum Hall states and quantum string liquids are examples of topologically ordered phases.
The low energy effective theory of topological phases happen to be topological quantum field theory. In nature, topological quantum field theory always appears as the
low energy effective theory of topological phase of quantum spins, bonsons, and/or electrons, etc. By definition topological phase is always a quantum phase of
quantum spins, bonsons, electrons, etc. ie topological phase is always a quantum
state of matter.