Ryan- Your question is so fundamental that I've entered it as a stand-alone rather than just another follow up on a long thread. You're getting close to philosophy, which is my excuse for why the answer will not be very satisfying.
Let's start with the less philosophical side. The electron cloud has a density which varies from place to place and can vary over time. There are all sorts of different states possible, and the way that that density varies in space is different for different states. The simplest states in atoms are spherically symmetrical balls, densest right in the middle and smoothly falling off farther away. Some other cloud shapes look more like figure-eights, with no density at the center and two big lobes opposite each other.
For an electron state that has a single value of energy, the shape of the cloud doesn't change over time, although there's an infinite variety of possible shapes. For states that have not only a range of positions (as they all do) but also a range of energies, the shape of the cloud changes over time. To understand how it changes one needs a more complete description of the state, going beyond the cloud density to include something called the wave function, a complex number whose absolute square gives the density.
For some physical processes, these clouds behave as if the mass and charge are spread out throughout the cloud, proportional to the density. However, in processes where different large-scale things would happen if the charge were concentrated in one part or another of the cloud, something very weird occurs. Say that some high energy photon (which you can picture as another cloud, just more tightly bunched) comes in and bounces off the electron. Then it gets detected on some screen, where all sorts of large-scale visible events show where it hits. It doesn't act like it bounced off that smooth cloud. Instead it acts like it bounced off an electron concentrated in some small part of the cloud. However, which part is not predictable, i.e. it's chancy. The odds of seeing it act as if the electron were concentrated in some region are proportional to the density in that region.
Now when I say "chancy" you may think that means that the electron really was somewhere specific and we just couldn't keep track of where. However, beautiful reasoning (Bell's Inequalities) shows that any such picture of quantum events gives predictions which contradict actual results.
How that initial spread-out cloud gets converted to a result which looks like what you'd get from a much smaller cloud randomly located in it is the next question. It could conceivably be answered in our lifetime. Some people think they know the answer, although they of course don't agree with each other. You can follow that whole issue under the name "quantum measurement". It may be our greatest mystery.
BTW, you may noticed that all I've talked about is various things that can be seen and the mathematical pictures for them. I haven't tried to give a verbal answer to "what is an electron?" I'm not sure that questions like that have answers.
(published on 11/05/2010)