Interaction with matter changes the neutrino mixing and effective mass splitting in a way that depends on the mass hierarchy. Consequently, results of oscillations and flavor conversion are different for the two hierarchies.
Sensitivity to the mass hierarchy appears whenever the matter effect on the 1-3 mixing and mass splitting becomes substantial. This happens in supernovae in large energy range, and in the matter of the Earth.
The Earth density profile is a multi-layer medium where the resonance enhancement of oscillations as well as the parametric enhancement of oscillations occur. The enhancement is realized in neutrino (antineutrino) channels for normal (inverted) mass hierarchy.
Multi-megaton scale under ice (water) atmospheric neutrino detectors with low energy threshold (2-3 GeV) may establish mass hierarchy with (3-10)σ confidence level in few years.
The main challenges of these experiments are flavor identification of events, accuracy of measurements of energies and directions, systematics, degeneracy of parameters.
There are various ways to improve the sensitivity to the hierarchy. This includes in particular, consideration of the v_e events, reconstruction of the cascade direction, etcetera. Inelasticity measurements will allow to increase significance of the hierarchy identification by 20-50%.
Detection of a neutrino burst from relatively close supernovae (which may occur any time) may resolve the issue of neutrino mass hierarchy.
Alexei Yu. Smirnov, "Neutrino Mass Hierarchy And Matter Effects", proceedings of the XV international workshop on "Neutrino Telescopes, ed. M.Mezzetto and G.Collazuol, Venezia, March 2013.

I quoted the summary of Smirnov's proceedings because I found it very clear. Of course the physics of neutrino mass effects and the issue of the neutrino mass hierarchy is a tough concept to explain in a blog post - I will make an attempt elsewhere if I have a chance.