There's a distinction to be made between two notions: foliations
and distributions. A distribution is the data, at each point m
of M, of a subspace of Tm(M). These subspaces are all of the
same dimension (say r), and depend smoothly on the point m,
which means that they are generated by r smooth vector fields. A
foliation is a partition of the manifold into (not necessarily
closed) submanifolds, such that, locally, this partition looks
like the standard decomposition of ℝn into translates of ℝd. Ok,
there's a caveat in my description since a same leaf could come
infinitely many often in the neighborhood of a given point m.
Anyways... I'm assuming that you know what a foliation is.
Foliations of M form a subset of distributions on M. The
Frobenius integrability criterion (mentioned by Tom in him
remark) states that a distribution D comes from a foliation iff
for any vector fields v and w tangent to D, their Lie bracket is
again tangent to D. It turns out that that criterion is always
satisfied for one-dimensio