A distribution of tangent vectors is called integrable if the Lie bracket of its sections is involutive, i.e. if $X,Y: M\to W\subset T M$ are two sections (vector fields belonging to the distribution) of $W\to M$ then the bracket $[X,Y]$ of these vector fields is also a section of $W$: $[X,Y]\in W$.

More generally, if a vector bundle is equipped with the structure of a Lie algebroid, then a distribution of subspaces is integrable if its sections are closed under the given Lie bracket. This reduces to the previous case for the tangent Lie algebroid. Hence integrable distributions are sub-Lie algebroids.

Properties

A basic result on integrability is the Frobenius theorem (Wikipedia) which relates involutivity to integrability in the sense of partial differential equations. Examples include complex analytic manifolds which correspond exactly to complex manifolds with an integrable almost complex structure. Courant algebroids are a quite general tool to express the integrability of geometric structure that include these as special cases.