Motivic homotopy theory is the homotopy theory for algebraic varieties and, more generally, for Grothendieck's schemes which is based on the analogy between the affine line and the unit interval. Ultimately, motivic homotopy theory is expected to provide techniques to solve problems in algebraic geometry such as various standard conjectures on algebraic cycles, Beilinson-Soule vanishing and rigidity conjectures, the Bloch-Kato conjecture etc. Unlike many other approaches tried in the past decades, motivic homotopy theory aims to understand the category of algebraic varieties internally, i.e. without explicit use of such constructions as Betti cohomology or Hodge structures which require external topological or analytic tools.
In its current state, motivic homotopy theory has as its primary objects of study categories of three types: the unstable A1-homotopy categories, the stable A1-homotopy categories and the triangulated categories of motives. Each of these types consists of a family of categories depending on the base scheme and in the two latter cases on the ring of coefficients.
It's only natural that motivic homotopy theory will have applications in quantum gravity. Perhaps there are immediate applications through the study of twistor string scattering amplitudes, which already make use of mixed Tate motives.