An integrable hierarchy is another name for a system of commuting Hamiltonian flows. The word "hierarchy" is used because a countably infinite number of commuting flows is obtained recursively.
[For the definition of a commuting flow, see for example the first part of this MO question.]
They arise from integrals of motion which are in involution (meaning that the Poisson bracket of any pair vanishes).
Commuting flows are useful, because they can be solved by the inverse scattering transform technique.
For an introduction from a mathematical perspective, see for example Introduction to integrable systems: open Toda lattice, KP, and KdV-hierarchies.