An integrable hierarchy is another name for a system of commuting Hamiltonian flows.    
<sub>[For the definition of a commuting flow, see for example the first part of this <A HREF="https://mathoverflow.net/questions/98534/when-do-commuting-hamiltonian-flows-have-commuting-generators">MO question.</A>]</sub> 
   
They arise from <A HREF="https://en.wikipedia.org/wiki/Constant_of_motion#Integral_of_motion">integrals of motion</A> which are in involution (meaning that the <A HREF="https://en.wikipedia.org/wiki/Poisson_bracket">Poisson bracket</A> of any pair vanishes).    

Commuting flows are useful, because they can be solved by the <A HREF="https://en.wikipedia.org/wiki/Inverse_scattering_transform">inverse scattering transform</A> technique.

For an introduction from a mathematical perspective, see for example <A HREF="http://gokovagt.org/proceedings/2009/ggt09-shapiro.pdf">Introduction to integrable systems: open Toda lattice, KP, and KdV-hierarchies.</A>