Discerning the best model for a problem
Unfortunately, there are many possible levels of description for a problem; choosing one among them is strictly dependent on your aims and on the tools available to you. In general, there are three possible approaches: analytical, numerical and observational. Each one is characterized by pro and cons. When you need to build a model, there are many possible decisions that must be made. For instance, you may decide among:


specific - general 
model estimation - first-principles models
numerical - analytical 
stochastic -deterministic 
microscopic - macroscopic 
discrete - continuous 
qualitative - quantitative


and so on. You must understand that, in general, there are no rigorous rules of thumb when making these choices. As pointed out by @Suresh, this a matter of experience, requires a careful understanding of the domain, and some inspired guessing as well. The good news is that once you have made your choices, then there are rigorous ways to proceed. Probably, the best model overall for a given problem is the one given by the Minimum Description Length Principle (essentially the Occam's razor): the best model is the smallest one. But determining it is beyond hope. So, the best model is almost always, the one that works best for you, given your explicit and even implicit assumptions.

That said, analytical and numerical approaches allows you to explore the behavior of a model, but the assumption here is that the underlying model is known. It's up to you to derive this model. Observational approaches allows you to derive a model from the experimentally observed data. 

Analytical models may lead to a closed-form solution; they are of great importance, given their power: if and when an analytical model may be successfully applied, then you will be able to deduce almost everything there is to know about a system. Their major drawback is limited applicability, since much of the world is simply too complicated to be described this way. Examples include ODE, PDE, difference equations, variational calculus and stochastic processes.

Numerical models are required when the equations in an analytical model are too complex to be solved exactly in closed-form. This usually happens as soon as we move too far from linearity to nonlinearity. It's important to remember that numerical and analytical solutions are complementary approaches rather than exclusive alternatives. Examples include Runge-Kutta methods, finite element methods, cellular automata and lattice gases.

Observational models can come from introspection or observation (or both).In some cases you may want to infer a model from experimentally measured data.This model may then be used to characterize and classify the data at hand, or to generalize from the measurements to predict the outcome of new observations etc. Examples include function fitting, SVD, Fourier and Wavelets transforms, neural networks, genetic algorithms, Expectation-Maximization algorithms, Wiener and Kalman filters, HMM, and time series. 

A very good introductory book you may want to read is The nature of mathematical modeling. Of course, given the huge amount of topics covered, do not expect a detailed treatment.