• mathematical modeling of phenomena that can be represented by systems of closed-form equations
• statistical modeling of phenomena that can be represented by distributions (the Central Limit Theorem, which you can find in any statistics textbook, will give a general explanation for the conditions under which we can pretend a sample of data is just like a normal distribution)
• solid modeling of structures or components, to allow us to visualize spatial relationships
• guesses (and sometimes even wild-ass guesses) for representations of phenomena we're just not sure about :-)

Always make sure of your assumptions. We model and simulate design trade-offs, hoping to understand potential real-world decisions before we actually MAKE them. And these real-world decisions involve dynamic behavior and randomness. Consider:

• performance optimization
• design corrections
• virtual environments; substitution for human capabilities (this use of simulation has applications in training and in -- ahem -- entertainment)
• implications of change over time; prediction of future behavior
• results of experimentation
• effects of events, whether expected or not; discovery (!)
• validation, verification, calibration (some call this last term "baselining"), formalization
• INSIGHT -- sometimes we just want to understand how things work

Simulation as engineers understand it often (if not usually) involves using computers to observe the behavior of changing systems. Because those changes are usually time-based, you'll often find simulations to be based on dynamic equations, such as the equations of motion of a vibratory system.

Here's a very simple simulation of 1/4 of an automobile suspension. (Without the math.) Can you see from this diagram what kinds of information the simulation can give you?

Here are some behaviors we create math models for as a matter of course:

1. Visible relationships between two variables. Especially when the relationship is linear. :-) The typical strategy is to eye-ball a line through that data, though we might use statistical methods (e.g. least squares) to fit something more precise. In many cases, it's OK to remember that "better is the enemy of good enough," and just eye-ball the fit. But if you do that, do it twice just to make sure of the relationship.
2. Events, or the chance of some event occurring (or not occurring) within a certain period. The precise strategy here involves an exponential density function. That density describes the distribution of durations between independent but more-or-less periodic events.
3. Random numbers. Most of us use a subroutine or canned function based on a pseudo-random binary sequence, generally one with a VERY LONG period so we never see the sequence repeat. :-)
4. Dynamic behavior. This is to say that the simulation is being excited by some input or internal characteristic changing in time.
5. Sorting and searching. I haven't had to include this in a dynamic simulation often. Thank GOD. LOL (These actions take a LONG time, relative to the others listed here.)

The simulation results are also going to be sensitive to

• how many times we run it
• how long we run it
• what sort of precision we use (though we have to remember that the more precision, the longer it takes to run -- and that this relationship isn't usually linear)
• what sort of output we require (animation will take longer than simple numerical output, and it'll be less transportable; but it WILL be easier to understand)
• the value we place on, well, sensitivity analysis

Limiting factors of simulation as a learning tool are:

• New users have to have some experience and a context.
• Not everybody can extract meaning from this exercise.
• Abstraction from simulations is easier in the classroom than in computer labs.

References

The Society for Computer Simulation
Arsham, H. System Simulation: The Shortest Route to Applications.
-- This site includes sample programs and nomenclature!
O'Haver, T. Simulations and Computer Models in the Classroom.
Plum, G. F. The Uses of Simulation.
Cool animation of automobile suspension simulation -- I've seldom seen such a detailed animated GIF, even at howstuffworks.com!