What are Einstein constraint equations? Yesterday, I met Einstein constraint equations in a thesis? I failed to understand them. Do they have physical meaning? And what do they "constrain"?
 A: Constraint equations, in the sense of the "Einstein constraint equations", arise when you try to write out an initial value formulation of GR.  The idea behind an initial value formulation, in general, is to show that if I hand you a set of data about the fields everywhere in space at some initial moment in time, that you can always find a solution to the equations of motion whose values matched the initial data at that initial moment.  This is similar in spirit to providing initial conditions for a plain old Newtonian point mass:  if I tell you the initial position and the initial velocity of a particle moving in a potential, then there's a unique solution to Newton's Second Law for the rest of time which satisfies these initial conditions.
So what are constraints?  Well, it turns out that in some (many?) field theories, you can't just pick any set of initial conditions you want.  Before we go to GR, let's deal with an easier example of this:  electrodynamics.  If we write out the equations for the scalar & vector potential in Lorenz gauge, in the absence of charge, we have
\begin{align*}
\Box^2 V &= 0 & \Box^2 \vec{A} &= 0 & \frac{\partial V}{\partial t} + \nabla \cdot \vec{A} &= 0.
\end{align*}
Now, the first two of these equations are just the wave equation;  and we know that if you tell me the initial amplitude of a wave and its initial rate of change at every point in space (i.e., $\psi(\vec{r}, t_0)$ and $\dot{\psi}(\vec{r}, t_0)$), then I can find a solution for $\psi(t)$ that satisfies these initial conditions.  So you might think that if I hand you $V(\vec{r}, t_0)$, $\dot{V}(\vec{r}, t_0)$, $\vec{A}(\vec{r}, t_0)$, and $\dot{\vec{A}}(\vec{r}, t_0)$, then you can always find a solution to the above equations.  Right?
Wrong, of course.  Because we can't freely specify our initial data;  they must obey an initial-value constraint.  In this case, the Lorenz gauge condition demands that $\dot{V}(\vec{r}, t_0)$ is exactly equal to $-(\nabla \cdot \vec{A})(\vec{r}, t_0)$.  However, so long as you hand me a set of initial data that satisfies this constraint initially, then the solutions of the wave equation that you find will satisfy this constraint for all time.
So what does this all have to do with GR?  Well, when we try to do the same thing for GR, we first have to split up our spacetime into a foliation (i.e., a set of 3D spacelike hypersurfaces that are parametrized by a time coordinate $t$.)  Einstein's equations can then be entirely rewritten in terms of $h_{ab}(t)$, the induced metric on the hypersurfaces;  and $K_{ab}(t)$, the extrinsic curvature of the spacelike hypersurfaces.  (Morally speaking, $K_{ab}$ can be thought of as the time derivative of $h_{ab}$.)  But, as in the case of electrodynamics, you can't just write down an arbitrary $h_{ab}(t_0)$ and $K_{ab}(t_0)$ on your initial hypersurface;  they are constrained relative to each other, by the equations
$$
D_b K^b {}_a - D_a K^b {}_b = 0,  \qquad  {}^{(3)}R + (K^a {}_a)^2 - K_{ab} K^{ab} = 0,
$$
where $D_a$ is the covariant derivative operator on the initial hypersurface and ${}^{(3)}R$ is its associated Ricci scalar.  This is analogous to how $\dot{V}$ and $\vec{A}$ couldn't be freely specified at $t_0$, but rather were related to each other.
I'm glossing over a lot of detail here, but that's the basic gist of it.  The best reference I know of for this subject is Wald's General Relativity, in Chapter 10;  the notions of constraints in classical field theory are also covered quite nicely in the first section of Dirac's Lectures on Quantum Mechanics (not to be confused with his Principles of Quantum Mechanics.)
