What is the definition of correlation length for the Ising model? The correlation length $\xi$ is related to critical temperature $T_c$ as 
$$
\xi\sim|T-T_{c}|{}^{-\nu},
$$
where $\nu$ is the critical exponent. 


*

*Is this the formal definition of correlation length? If not, what is the formal definition of correlation length (for phase transition in the Ising model)?

*Can you give a physical understanding of correlation length?

 A: The two-dimensional square-lattice Ising model, which is a simplified model of reality, exhibits a phase transition. Onsager showed that there is a specific temperature, called the Curie temperature or critical temperature $T_c$, below which the system shows ferromagnetic long-range order. Above it, it is paramagnetic and is disordered.
At zero temperature, every spin is aligned in either +1 (or -1) direction. When we increase the temperature, keeping below $T_c$, some spin of starts orienting themselves in the opposite direction. The typical length scale of cluster formation is called correlation length, $\xi$, and it grows as we increase the temperature and diverges at $T_c$. If we go beyond $T_c$, the correlation length starts decreasing, and at infinite temperature, it becomes zero.

2-dimensional Ising model simulation on 100x100 lattice. Left to right and top to bottom, the temperature is increasing. At equilibrium, when $T<T_c$, typical configurations in the + phase look like a ''sea'' of +1 spins with ''islands'' of -1 spins. For larger lattice sizes, the ''islands'' have ''lakes'' of +1 spins. In this picture, +1 spins are in black, and -1 spins are in white. Each connected white object is a cluster.
Formally:
The two-point correlation function is defined as
\begin{equation}\label{Gamma}
 \Gamma(i-j)=\langle S_iS_j\rangle-\langle S_i\rangle\langle S_j\rangle
\end{equation}
The correlation length, $\xi(T)$ is characteristic length at which the value of correlation function $\Gamma(i)$ has decayed to $e^{-1}$:
$$\Gamma(i)\sim \exp\Bigg(\frac{|i|}{\xi(T)}\Bigg)$$
And
\begin{equation}\label{correlationlength}
\xi(T)\sim |T-T_c|^{-\nu}
\end{equation}
For $d=2$, we have $\nu=1$.
A: That is not a definition of correlation length. (It is a definition of the critical exponent.)
The correlation length is defined in terms of the 2-point correlation function of spin observables.  Pick points $x$ and $y$ on the lattice, and consider the expectation value $\langle s(x) s(y) \rangle$ of the product of the spin observable at $x$ and the spin observable at $y$.  This quantity tells you how strongly correlated the spin at $x$ and the spin at $y$ are, as a function of the temperature, coupling constant, and the distance between $x$ and $y$.  If $T > T_c$, then the correlation function dies off exponentially fast in $|x-y|$.
$\langle s(x) s(y) \rangle \sim e^{-\frac{|x-y|}{\xi(T)}}$
The correlation length is, by definition, the constant (in $x$ and $y$, but not in $T$) which tells you how fast the correlation function vanishes.  
A: Just a small addition to what user1504 said: the correlation length can be defined for $T<T_c$ as well, so that
$\big\langle\big(s(x)-\langle s(x) \rangle \big)\big(s(y)-\langle s(y) \rangle \big)\big\rangle=e^{-\frac{|x-y|}{\xi}}$
A: Since technical derivation and explanation of correlation length have already been discussed in detail, I would rather share my understanding of this subject matter.
The notion of correlation length is quite general in the study of thermal or quantum phase transition. It is the only relevant length scale near the critical point.
Let's think about a magnetic system. Usually, nearby spins tend to be correlated. Away from the critical point, $T \neq T_c$, their correlation extends to a certain distance $\xi$, called the correlation length. This is
the typical size of the regions in which the spins assume the same value, as shown below

Where the size of the magnetic domain is given by the correlation length. Of course, one can make its definition more precise in terms of the asymptotic behavior of the correlation function, but the physical picture remains the same as corresponds to the diagram above.
