Types of quantum field thories
In the first order formulation of gravity the field of gravity is modeled by a connection with values in the Poincaré Lie algebra. Such a connection locally decomposes into a -Lie algebra valued 1-form called the vielbein and a so(d,1)-valued 1-form called the spin connection piece.
There is a constraint on this data. Usually the -connection is required to be the Levi-Civita connection corresponding to the pseudo-Riemannian metric encoded by the vielbein, hence the unique metric compatible connection with vanishing torsion but in general non-vanishing curvature.
But to some extent one may instead impose a dual constraint and demand the connection to be the Weitzenböck connection, which has vanishing curvature, but in general non-vanishing torsion. This formulation of gravity is called teleparallel gravity, because by flatness of the connection, its parallel transport is trivial and hence tangent vectors at different points of spacetime may consistently be compared without specifying a path between them, hence without first parallel transporting one to the other. This “parallelism at a distance” is what gives teleparallel gravity its name.
Notice that under some conditions a manifold being flat (i.e. admitting a flat connection on its tangent bundle) implies that it is a parallelizable manifold, hence that the tangent bundle is actually trivializable. Sufficient such condition is notably that the flat manifold is also simply connected, but this may be relaxed (Thorpe 65). Some authors may explicitly assume for teleparallel gravity that the tangent bundle is trivialized by the vielbein field (making it a framed manifold), though authors in the physics literature tend to be vague on this point.
At least without coupling to matter, the teleparallel formulation is locally classically equivalent to the ordinary formulation in first order formulation of gravity in that inserting in the Euler-Lagrange equations for the vielbein+Weitzenböck connection the expression of the latter in terms of the Levi-Civita connection one reaobtains the ordinary Einstein equations.
Despite this equivalence, the map of the formal field content of teleparallel gravity to its physical interpretation has a rather different feel to it than in the ordinary formulation. Where ordinarily the field of gravity does not quite affect the motion of bodies like an ordinary force in Newtonian physics, but instead by curvature causing non-trivial geodesics, in the teleparallel description the curvature does vanish just as in newtonian physics, and instead the torsion of the connection acts like a genuine Newtonian force.
Survey of basic ideas and issues are for instance
J. Pereira, Torsion and the description of the gravitational interaction (pdf slides)
R. Aldrovandi, J. G. Pereira, K. H. Vu, Gravitation without the equivalence principle (arXiv:gr-qc/0304106)
More discussion is in
The Schwarzschild spacetime in the context of teleparallel gravity is discussed in
The relation between flat tangent connections and parallelizablity is discussed in