A conformal structure on a manifold is the structure of a Riemannian metric modulo rescalings of the metric tensor by some real valued function on the manifold. The homomorphisms of conformal structures are called conformal transformations.
In terms of Cartan geometry conformal structure is expressed by conformal connections, conformal geometry is a special case of parabolic geometry and hence of Cartan geometry.
In the context of quantum field theory conformal structure underlies the formulation of conformal field theory. Due to the coincidence of the conformal group of Minkowski spacetime of dimension $d$ with the anti de Sitter group of anti de Sitter spacetime in dimension $d+1$ there is a close relation between certain conformal field theories and certain theories of gravity. This is the content of the AdS-CFT correspondence. This works most accurately in the context of supergeometry, hence for superconformal groups acting on the asymptotic boundary of super anti de Sitter spacetimes.
David Gu, Conformal structure, slides (pdf)
Boris Apanasov, Doubles of Atoroidal Manifolds, Their Conformal Uniformization and Deformations In: J. Ławrynowicz (ed.) Deformations of Mathematical Structures Springer, Dordrecht (1989) (doi:10.1007/978-94-009-2643-1_9)
Boris Apanasov, Conformal Geometry of Discrete Groups and Manifolds, De Gruyter Expositions in Mathematics, 32, De Gruyter (2000) (doi:10.1515/9783110808056)
Discussion of conformal structure as G-structure:
Discussion of conformal Cartan geometry (parabolic geometry) includes
Andreas Čap, Jan Slovák, sections 1.1.5, 1.6 of Parabolic Geometries I – Background and General Theory, AMS 2009
Felipe Leitner, part 1, section 6 of Applications of Cartan and Tractor Calculus to Conformal and CR-Geometry, 2007 (pdf)
Hega Baum, Andreas Juhl, Conformal Differential Geometry: Q-curvature and Conformal Holonomy, Oberwolfach Seminars, vol. 40, Birkhäuser, 2010, 165pp.
Andree Lischewski, section 2 of Conformal superalgebras via tractor calculus, Class.Quant.Grav. 32 (2015) 015020 (spire, arXiv:1408.2238)
Sean Curry, A. Rod Gover, An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity, 2014, (arXiv:1412.7559)
A survey of the field as of 2007 is in
Discussion with an eye towards combination with spin geometry is in
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