The notion of ending laminations was first developed by Thurston for Kleinian groups isomorphic to a surface group, and satisfying a condition called "geometric tameness". In this case (assuming for simplicity that there are no parabolic elements), he showed that each end of the manifold was either geometrically finite or totally degenerate. He first encountered totally degenerate ends in the study of cyclic covers of hyperbolic 3-manifolds fibering over the circle. In these examples, because of the translational symmetry, there is a sequence of geodesics of bounded length exiting each end. One could analyze this sequence of curves on a fixed surface embedded in the manifold, and see them converge to the stable or unstable lamination of the pseudo-Anosov monodromy of the mapping torus. So Thurston defined a geometrically tame end of a hyperbolic manifold to be an end with a sequence of embedded closed geodesics exiting the end which were isotopic to simple closed curves on the surface. He showed that these curves converge to a unique lamination on the surface, called the ending lamination (which is recurrent but not measured). Intuitively, this invariant measures where the "short" curves are on the surface as one exits the end. Thurston used these notions in the proof of his Double Limit Theorem, an important special case of the geometrization conjecture.
Bonahon developed Thurston's notion further, proving that a Kleinian surface group (more generally, indecomposable Kleinian group) is geometrically tame. He showed that any sequence of geodesics exiting the end converged to a unique geodesic current on the surfac, and then used some of Thurston's estimates to prove that in fact, this geodesic current had self-intersection number zero, and therefore was a lamination. He went on to prove that the end is geometrically tame using this result.
The notion of ending lamination was generalized by Canary in the compressible case when the end is topologically tame. Topological tameness was proved by myself and Calegari-Gabai, thus defining the ending lamination for general Kleinian groups. In the compressible case, the ending lamination must lie in the Masur domain.
Thurston showed that there is a compactification of Teichmuller space by measured laminations. Since geometrically finite Kleinian groups are parameterized by the Teichmuller spaces of their conformal boundaries at infinity, it might be natural to assume that one could parameterize limits of geometrically finite Kleinian groups by measured laminations. However, this is not quite the right notion: one needs to eliminate the measures. The ending laminations turn out to be the ends of the curve complex of the surface associated to the end of the hyperbolic 3-manifold. The content of the ending lamination theorem states that the ending laminations of degenerate ends together with the conformal structure of the geometrically finite ends uniquely determines the Kleinian group. The case with rank one parabolic present is a bit more complicated to state, but is a similar classification.