I haven't found very many great examples online…could someone explain a process of iteration to me in easy steps? I'm looking at this from a dynamics point of view if that helps any…

Dynamics is a pretty wide field, could you be more specific?

Iteration is a process normally used to converge on an answer. Sometimes, rather than hacking out the purely mathematical way of finding something (if a purely mathematical way even exists), sometimes we guess and check. If you have an idea of the range of your answer, sometimes you can plug in a, "guess," to an algorithm, and the algorithm will spit out an answer closer to what you are looking for. Replug THAT answer in, get another one even closer to your desired answer. Eventually, these answers will converge to one answer that you are looking for.

One such example of iteration used in dynamics for finding roots (using a Taylor series approximation) is the Newton-Raphson method.

The wiki article here actually does a pretty good job of explaining how it works. Read through that carefully, and you will begin to understand why iteration is used, and how it is implemented.

Finding frictional coefficients in piping systems. I'll look at that link sometime today.

If you're trying to do parameter estimation, least-squares fitting (linear or non-linear) is often used as well.  I use this technique pretty often when I have experimental data and a rough model I need to estimate parameters for.

There are still many things in electrical engeiiring that are most easily done by iteration,
Make a reasonable assumption about current in a transitor circuit, calculate values based on data sheet plots, alter inital number based on result, re-run calculations again.
When the numbers calculated show only a small change from iteration to iteration you are done.

Something as simple as the current though a Light Emitting Diode with (LED) a current limiting resistor works this way.
The voltage drop of the LED changes with the current, and the change in current alters the voltage drop though the resistor.
Estimate voltage drop on LED, compute current, look up LED voltage drop at that current, use THAT drop and recompute.
One or two cycles should get you withing a few percent.

It is faster than trying to curve fit an equation to the LED current-voltage curve, even with curve fitting software like Easycurve.

Of course, as the current and voltage operating point change, power dissipation changes.  As power dissipation changes, so does the temperature.  As the temperature changes, the current-voltage curve for the LED changes.  And on and on it goes.

Right, but all those factors should converge (they do in the physical device, after all).  At the end of the day, iteration is just a technique for root-finding of some arbitrary function.  You can throw as much or as little complexity into the model function as you need.