I'm looking for some simple explanations of the various types of filters such as:

What do these terms mean: Butterworth, elliptical, poles and zeros, etc., etc.

 

A Butterworth response is derived from a requirement for maximal flatness in the passband and transition band. It yields a particular rolloff characteristic in terms of how fast the response rolls off in the transition band. An elliptical filters is derived from a different set of requirements including a very narrow transition band with a very steep rolloff. This is achieved at the expense of ripple in the passband and in the stopband. All of the filters you will ever see besides all the ones that you won't start with requirements for attenuation and ripple in the passband, transition band, and stop band.

All filter responses can be related to a plot of the poles and zeros of the transfer function. When plotted in the complex plane a pole is located where the denominator of the transfer function goes to 0. A zero is located where the numerator of the transfer function goes to 0. Real poles occur on the real axis. If they are on the negative real axis then the system response does NOT diverge to infinity. If they are on the positive real axis then the system response will diverge to infinity and the system will destroy itself. Complex poles will always occur in conjugate pairs. Zeros can also be located on the real axis or in conjugate pairs.

https://en.wikipedia.org/wiki/Transfer_function
https://www.dsprelated.com/freebooks/sasp/Lowpass_Filter_Design_Specifications.html
http://web.mit.edu/2.14/www/Handouts/PoleZero.pdf

 

So as a practical matter (without all the mathematical theory) what is the difference between all these filters?

 

The main differences are the ripple in the passband (and stopband for the elliptical), the abruptness of the rolloff, the phase response, and the pulse response.
Which of those are most important in your application determines which filter type you use.
Here's a discussion of the differences between the common filter types.

 

I'm looking for some simple explanations of the various types of filters such as:

What do these terms mean: Butterworth, elliptical, poles and zeros, etc., etc.

Hi,

Poles and zeros are considered special frequencies that have a relationship to the frequency response of the filter and a special relationship to the transfer function. Poles are frequencies that force the denominator of the transfer function to infinity, and zeros are frequencies that force the numerator of the transfer function to zero. When these frequencies occur, the filter response exhibits a certain shape depending on whether it is a pole frequency or a zero frequency.
For a simple pole the response becomes 3db down and starts to drop 20db per decade and for a simple zero it starts to increase at 20 db per decade. That was Bode's original idea, to use straight line approximations to approximate the response of the filter which can help in both analysis and design. For a simple zero the response starts to increase by 20db per decade.

For a simple example with the simple transfer function:
H(s)=1/(s*R*C+1)

if we just look at the denominator:
s*R*C+1

we find that this goes to zero when s=-1/RC so w=1/RC is a pole frequency so the response will be down by 3db and will be deceasing by 20db per decade.

That transfer function must be a low pass filter because after a certain point the frequency response decreases.

There are other considerations, but that's the basic idea of a pole. If a zero is encountered, the response starts to increase so it may appear to cancel the effect of the pole at a later frequency.