What are correlated magnetic moments? My book has the following sentence and I don't understand what correlation or lack of correlation means: 

At high temperature the magnetic moments of adjacent
  atoms are uncorrelated (to maximize the entropy) so the crystal has no net magnetic moment.

The book is touching on second-order phase transitions and it's describing how magnetic transitions are an example of such 2nd order phase transitions.
 A: Correlation between two variables (or objects) is, very simply put, how much a change in one variable affects or determines a change in the other. Replacing variables with spins, highly correlated spins would mean that, due to some interactions between them, a change in the direction of one spin will cause a change in the direction of the spin it is correlated with. Thus a high degree of correlation would imply that the behaviour of each spin is governed by its neighbours and overall, they tend to cooperate and point in specific directions rather than randomly, giving long-range ordering. 
At high temperatures, the energy scale of this correlation or 'talking' between spins is much weaker than the thermal energy scale. So the directions of the spins are thermally fluctuating and point randomly rather than determine the position of the neighbouring spin. So there is no more long range ordering and the system shows no net magnetic moment. 
User17338, hope this helps. Others, please correct me if I am wrong. :)
A: It means that the directions of the magnetic moment of neighboring atoms is statistically independent.  At high temperature the thermal energy is greater than the magnetic energy; the resulting thermal fluctuations cause any material to become paramagnetic.  As you lower the temperature these thermal fluctuations are reduced.  Ferromagnetics and anti-ferromagnetics have long range order below the temperature at which thermal fluctuations are suppressed.  This is a big topic of course, but I hope this little bit helps.  The following book is excellent.
“Magnetism in condensed matter” by Stephen Blundell
Oxford masters series; Oxford University Press, 2001; ISBN: 0198505914
A: Correlation means how likely they are pointing in the same direction together. More qualitatively, it is defined as $\langle s_i s_j \rangle$ between sites.
In non-interacting system, all spins are free and independent of each other. For each pair, the probability of pointing in same direction or opposite direction is equal, so the correlation is zero.
However, in interacting system like Ising model describing ferromagnetic, the system has lower energy when spins align with each other. Therefore, spins in system are pointing in the same direction at low temperature, so correlation between spins is high. In contrast, at high temperature, the thermal noise is large enough that they can point in random direction, so correlation is low.
More quantitatively, considering 4 adjacent atoms with fixed spin up, and the interaction energy is $\Delta E$. Then the probability of the center atom in down spin ($\downarrow$) and up spin ($\uparrow$) are proportional to:
$$p_\downarrow \propto e^{+4\beta \Delta E}$$
$$p_\uparrow \propto e^{-4\beta \Delta E}$$
respectively. Hence, at high temperature, $T\to\infty$, so $\beta\to 0$ and $p_\uparrow=p_\downarrow=0.5$, i.e. they are completely uncorrelated. Similarly, at low very temperature, $p_\uparrow=1$, i.e. it always align with its neighbour and so they are strongly correlated.
