It is well-known that in order to define a relavistic quantum-field theory containing fermions on a general manifold $M$, the manifold $M$ needs to be equipped with a spin structure. The spin structure is a lift of the frame bundle (which is a principal $SO(n)$-bundle) into a principal $Spin(n)$ bundle, allowing us to define transport of spinor fields.
On the other hand, spin structures also seem to come up in condensed matter physics when studying topological phases of systems involving fermions. These systems are typically defined on a lattice, with no Lorentz-invariance or spin-statistics theorem. There are typically no spinor fields in sight, and certainly no need to define their transformation properties in $SO(n)$. So it is far from obvious why spin-structures would be important. Nevertheless, it is conjectured  that topological phases are classified by Spin-TQFTs (Topological quantum field theories over manifolds equipped with a spin structure), suggesting that the lattice models can still only be defined on spaces equipped with a spin structure. In particular models , this seems to be borne out (the spin structure enters via the need to choose a "Kasteleyn orientation" on the lattice). But I am looking for a more general explanation.