I've been reading through some of the literature on solutions of the Ising model, but I can't help but notice it doesn't provide that good a model for actual ferromagnetic systems. It seems that these models get a lot of attention and I'm just curious as to why? Also, why is an exact solution to the 3D Ising model so sought after, and what do people hope to learn from it? I'm a bit naive on the subject, so I was just hoping someone could help me get a better understanding.
Mainly because the 2D Ising model is one of the few models of a phase transition which is analytically solvable (and that only in the absence of a magnetic field). But you are right that it isn't a good model for ferromagnetism. The problem is that in two dimensions there is no phase transition with breaking of a continuous symmetry due to the Wagner Mermin theorem. In the case of the Ising model, a discrete symmetry is broken, that's why also in two dimensions a phase transition is observed. However it seems impossible to find an analytic solution in 3 dimensions. The importance of the Ising model lies in that the results of approximate methods of solution can be gauged and tested against the analytic solution.
I've also seen it show up in the context of QFT, so what's the explanation there? It doesn't seem to model any sort of field I can think of so why do they use it?