A typical example: the set of Weyl structures in conformal geometry.
Background: In contrast to Riemannian manifolds which carry a unique torsion-free metric connection, a conformal manifold has many torsion-free connections preserving the conformal structure (they are in 1-1 correspondence with connections on the weight bundle, and thus form an affine space directed by the space of 1-forms). Any such connection is called a Weyl structure.
Application: Using Weyl structures one might define so-called Gauduchon gauge on conformal Hermitian manifolds. This is the unique metric $g$ in the conformal class whose Lee form is $g$-co-closed. Gauduchon gauges have important applications in geometry (e.g. Kobayashi-Hitchin correspondence, see http://www.cmi.univ-mrs.fr/~teleman/documents/universal-05.pdf).