Short answers, pulled from the story.
The torsion tensor is a bilinear map associated to any affine connection on a manifold. It takes two input tangent vectors and produces an output vector that represents the displacement of a developed curve when a tangent space is rolled along an infinitesimal parallelogram. It is skew-symmetric in its inputs.
Two affine connections that share the same family of affinely parametrized geodesics differ only by torsion. Given any affine connection, there is a unique torsion-free connection that generates the same geodesics; the difference between the two is called the contorsion tensor. Selecting this torsion-free connection is called absorption of torsion.
Absorption of torsion is the process of choosing the unique torsion-free connection subordinate to a given family of parametrized geodesics. It is one of the central stages of Cartan's equivalence method, used to determine when two geometric structures are equivalent. It generalises the fundamental theorem of Riemannian geometry to non-metric affine connections.
When a small closed parallelogram is developed into the tangent space using a connection with non-zero torsion, the developed curve fails to close: it ends at a point displaced from its start. That displacement vector equals the torsion tensor applied to the parallelogram's side vectors, and it is directly analogous to the Burgers vector that quantifies dislocations in crystallography.
In elasticity theory, torsion describes the difference between the energy-minimizing helical configuration of a vine modelled as twisted elastic filaments and its length-maximizing geodesic configuration. In fluid dynamics, torsion is associated with vortex lines, and the Bianchi identities of a connection with torsion yield equations describing an equilibrium continuous medium with moment density.
The Bianchi identities are two relations every affine connection satisfies. The first links the cyclic sum of the curvature tensor's action on three vector fields to a cyclic expression involving the torsion. The second links the covariant derivative of the curvature to a similar torsion-dependent cyclic sum. These identities constrain which combinations of curvature and torsion a connection can possess.