A T-fold is a space that locally looks like a Riemannian manifold equipped with a B-field, but is glued together from these not just by diffeomorphisms but also by T-duality transformations.
The idea is that a T-fold is a target space for a string sigma-model that is only locally a Riemannian manifold but globally a more general kind of geometry. In the literature sometimes the term non-geometric backgrounds is used for such “generalized geometric” backgrounds.
The idea was originally introduced in
Further developments are in
Chris Hull, Doubled geometry and T-folds JHEP0707:080,2007 (arXiv:hep-th/0605149)
Chris Hull, Global Aspects of T-Duality, Gauged Sigma Models and T-Folds (arXiv:hep-th/0604178)
Aaron Bergman, Daniel Robbins, Ramond-Ramond Fields, Cohomology and Non-Geometric Fluxes (arXiv:0710.5158)
Dieter Lüst, Stefano Massai, Valentí Vall Camell, The monodromy of T-folds and T-fects (arXiv:1508.01193)
A precise global definition of T-folds as principal 2-bundles for the T-duality 2-group described in the nLab entry T-Duality and Differential K-Theory is given in