Poles and Zeros in English
In short, poles and zeros are a way of analyzing the stability of a feedback system.

I'll try not to get too math-heavy but I'm not sure how to explain without at least some math.

Here is the basic structure of a feedback system:



In this form there is no gain or compensation in the feedback path, it is placed entirely in the forward path, however, the feedback portion of more general systems can be transformed to look like this and analyzed in the same way.

The transfer function in the box is called \$L(s)\$ because the analysis is often done in the Laplace transform space. Laplace transforms are similar to fourier transforms, so you can think of L(s) as a frequency response. For example, a perfect low-pass filter has \$L(s) = 1\$ for \$s\$ less than the cutoff frequency, and \$L(s) = 0\$ above the cutoff frequency.

\$L(0)\$ is the DC gain of the system. For a feedback control system, a large DC gain is desirable because it reduces the steady-state tracking error of the system. 

Poles and Zeros

\$L(s)\$ is a complex-valued function. Usually the polar form \$A e^{i \theta}\$ is used; $A$ is the magnitude and \$\theta\$ is the phase. The magnitude of \$L(s)\$ is also called the gain.

The poles and zeros provide a convenient, fast way to think about the properties of \$L(s)\$. When doing a rough plot of \$L(s)\$, poles contribute -90&deg; of phase above the pole frequency and cause the magnitude to "roll off" (decrease). Zeros do the opposite &mdash; they contribute +90&deg; of phase and the magnitude increases. This will probably make a lot more sense looking at the pictures and the "Rules for hand-made Bode plot" section of http://en.wikipedia.org/wiki/Bode_plot.

For a system to be stable, the magnitude of \$L(s)\$ needs to drop below unity before (at a lower frequency) the phase reaches -180&deg;. Typically some margin is required here; "gain margin" and "phase margin" are two ways of measuring how far \$L(s)\$ is from the (1, -180&deg;) point.

As a simple example, an op-amp might have \$L(s) = \frac{10^{6}}{s}\$. In this case there is a pole at zero and no zeros. As you would expect for an op-amp, there is a large DC gain. The gain drops off as the frequency increases from DC (due to the pole at zero). According to this model, the system cannot be unstable because the phase is never less than -90&deg;.

When reading an app note that talks about poles and zeros, you may need to figure out the general form of \$L(s)\$ for the system in question, or you may be able to draw some conclusions just from a list of poles and zeros. Adding either a pole or a zero to a system will change both the gain and phase margin; adding a pole and a zero together (at different frequencies, both below the -180&deg; crossover) will change the gain margin but not the phase margin. Adding two zeros and two poles can create a hump in \$L(s)\$ (think bandpass filter), without changing either the gain or the phase margin.

Hope this helps. In general I would expect datasheets and app notes to suggest values for the compensation components so that the user doesn't need to analyze the stability unless there are special requirements. If you have a specific part in mind that you're having trouble using and you post a link the datasheet, I might be able to offer something.