I think most of the time people know that reality is approximated but cannot, or will not bother to, try to estimate/quantify how bad the approximation actually is.

The number of parameters/variables/equations is often viewed as an indicator of model complexity (simplicity) - size related complexity. This is because it is easier to wrap ones head around a model that uses a smaller number of “symbols” to represent it. However, there is more to system/model complexity than its size. My opinion is that analytical approachability, computational feasibility and the nature of the model’s underlying abstractions all contribute to the complexity of a model.

A model is only truly faithful to reality if it is as complex as the real-world process it is simulating. In that case, it would usually be easier to just observe or experiment on the real-word process (which is what we usually do in medicine).

To be useful, the required outputs of the model should predict those of the real-world process sufficiently accurately, but using only inputs which are easily available or acquired through direct measurement. The internal workings of the model are hidden from the user, so complexity here is not important to the user. However, the cost of developing, maintaining and testing the model increases with increasing complexity.

In general, a model with few inputs will be simpler than one with more inputs. 

Nice point of view, James!

Nice point, Jimmy!

Perhaps a relative answer to your question is more realistic than an absolute one. In that case we think of two models with comparable performance and then ask which of them is simpler.

As for the general features, assuming the system being modelled is dynamic, features of DECREASING simplicity would be: static models (linear, then nonlinear), dynamic linear models, nonlinear models that are linear in the parameters, and nonlinear models that are not linear in the parameters. To these classes we might add time-invariant, and time-variant.

Having established those general features of the model, then the number of parameters and number of inputs, as pointed out by Drs. Derbal and Tattersall, are good criteria. I particularly like the suggestion by Dr. Derbal that analytical approachability is an interesting criterion.

The practial difficulty with this issue is that: 1) there is no general threshold above which a model is no longer simple, 2) we usually have to trade simplicity with performance. So in the end, it seems that simplicity must be addressed in relative terms... In fact, that is what is behind Ocam's razor principle. 

I believe we should distinguish between conceptual complexity and computational complexity (or computational cost). A model can be conceptually extremely simple in terms of required inputs, but computationally very complex  (or computationally very demanding) for example because a PDE is being solved numerically on a very fine mesh. I would tend to consider such a model a simple one, provided that reliable  methods for its numerical solution exist. For the same reason, I would also tend to relax the requirement that, for a model to be considered simple, an analytic solution must exist. It is clear that this adds another layer of 'complexity', but modern scientific computing provides in my opinion a sufficiently wide array of tools to keep this complexity under control. I may be wrong, but my feeling is that some models are deemed too complex only because they entail more sophisticated numerics than what was the standard 20-30 years ago, as in the saying 'Never solve a PDE if you can solve an ODE instead'.