You keep trying to understand it "more electrically", but the problem is that pole dominance, and the pole concept itself, is strictly related to Laplace Transform (LT) theory, which is a strong mathematical theory.
Poles are not an "electrical concept" in itself. It is a concept that can be applied to any linear system, or any system amenable to be linearized usefully for the application at hand. They can be electrical systems, but also mechanical, thermal or systems in whatever field of application you may imagine where the LT techniques can be applied.
Chu already gave you a very simplified view of the concept in the time domain for an electrical system. I'll try to dumb down the things some more, but keep in mind that if you don't want to grok some math, it is really not possible to understand the concept in all its depth.
First of all, poles are related to how fast a system can react when it is excited, and the reaction time is related to how the system can store and release energy during and after the excitation.
An (idealized) system which cannot store energy has no reaction time, it reacts immediately to any stimulus it is subjected to (think of a pure resistive circuit, where resistors are ideal). Reaction time for electrical circuits, therefore, depends on elements that can store energy: reactive elements, that is capacitors and inductors (if we keep the things simple and don't consider mutual induction and/or distributed circuits).
Grossly simplifying, each independent reactive element contributes a pole to the circuit, and the number of poles of a system is called the order of the system. So a circuit with one cap and no inductors is a 1st order system. A circuit with a cap and an inductor is a 2nd order system (or two independent caps), and so on.
What does it mean "independent" element? Too complex for you, if you don't want to grok the math, so let's skip it. Very broadly, it means that the level of energy storage of one element is completely independent from the level of the other elements.
OK, now what do the poles do to a system? They describe the "reaction time***s***" of the system. Note the plural! A system that is excited momentarily (imagine a fast electrical impulse) reacts with a response that evolve in time and then dies off (if the system is stable).
This "impulse response" is made up of different components, each of which decays with a different time constant. Every pole contributes a different component with a different time constant. The time constant is a measure of how fast a component decays: the bigger the time constant, the slower the component decay (usually a component is considered zero after 5 time constants have elapsed, actually the decay lasts forever theoretically since it is exponential -- see Chu's answer for a simple math expression).
The impulse response of a stable system dies off after a while and the time it takes to die off gives you an estimate of the reaction time of the system: the time a system needs to stabilize into a new condition when the inputs are varied.
A dominant pole is a pole whose time constant is much "slower", i.e. bigger, than all the other time constants of the circuit, therefore the corresponding component is still observable after all the other, faster decaying, components have died off.
In other words, a system with a dominant pole behaves approximately like a 1st order system, i.e. a system that has a single pole.
In the end, the concept of pole dominance is a way to simplify a system: if a dominant pole exists, with some caveats, the system can be thought as a 1st order system, i.e. the simplest system imaginable that has a non-trivial time evolution.
The problem is that not every system has a dominant pole, so it can't be simplified that way. A technique used sometimes to simplify things is to add a compensating network in the circuit, i.e. a network containing at least one reactive element, in order to introduce a dominant pole in the system and thus allowing the simplification.
Of course, if you want to introduce a new pole in the system that has to be dominant, its time constant must be bigger of any existing pole (unless you do pole-zero cancellation, but that is a still nastier and more advanced technique), therefore you end-up slowing down the system. So it's often a kind of trade-off: you slow down a fast system in order to make it simpler, i.e. more manageable by the rest of the control chain. In the process you lose something (e.g. fast response or gain) in order to gain something else (e.g. stability or accuracy).
Note that my answer contains lots of hand-waving and gross conceptual simplifications, but everything I've said can be justified and expanded mathematically to give the exact definition and interpretation of what is a pole and what is a dominant pole. The problem is, as I've already said, that you can't escape the heavy math then.