Hydrodynamics is an effective (emergent) theory that describes the long-time, long-distance (small frequency and wave-number) dynamics of most interacting many-body systems. The hydrodynamic limit is the limit $\omega\to 0$, $\vec{k}\to 0$. The hydrodynamic description is systematically improvable order by order in $\omega$ and $k$.
If the system is confined to a finite volume, then as $t\to\infty$ complete equilibration takes place, and hydrodynamics reduces to thermodynamics. Hydrodynamics is then a dynamic, time-dependent, generalization of thermodynamics. In the hydrodynamic limit systems are in approximate local equilibrium, but not necessarily in global equilibrium.
There are many kinds of many-body systems, and many hydrodynamic theories. These fall into classes, dependning on the symmetries, the dimensionality, and the number and type of conserved or quantities. Historically, the first system to be studied is the theory of non-relativistic many body systems (like water or air, which is why the theory is called hydrodynamics) that are governed by conserved particle number, energy, and momentum. The corresponding theory is called the Euler equation, and including the first correction in $k$ leads to the Navier-Stokes equation.
Hydrodynamic theories can be derived from more microscopic theories, like quantum field theory, many-body quantum mechanics, or classical many-particle physics. It is also possible to derive hierarchies of effective theories. For example, kinetic theory is an effective theory of particle distribution functions that can be derived from quantum or classical many-body theories. At long distances, kinetic theory reduces to hydrodynamics, and hydrodynamic theories and their parameters can be derived from kinetic theory. The basic equation of motion of kinetic theory is the Boltzmann equation, and derivation of hydrodynamics from the Boltzmann equation were studied by many people, for example Grad, Chapman, Enskog, etc.
