This means that an electron is not merely a particle whose properties are inherently "uncertain." Rather, those properties (e.g., the electron's position -- a particle property) don't even exist until the experimenter measures them via an experiment that demands a particle-like answer. Let's say it again: until that electron reaches the fluorescent screen and makes a flash of light, it doesn't have any position. This is why the uncertainty principle is often called the "indeterminacy principle" instead: the properties of electrons, protons, photons, etc., are indeterminate (undefined), not merely unknown to us humans.
2.
All particles have wave properties, so all particles can produce diffraction patterns! Since electrons are nearly 2000 times less massive than neutrons, it's a lot easier to see their diffraction patterns, since their "de Broglie wavelengths" are much larger (for a given particle speed). But neutrons, sodium atoms, and airplanes all exhibit diffraction and other wave phenomena.
Just don't hold your breath waiting to see experimental evidence of airplane diffraction.
3.
According to Heisenberg's uncertainty principle, we can never know a particle's position and its momentum, at the same time, to arbitrary precision: the product of the uncertainties on these two "conjugate variables" can never be smaller than Planck's constant h divided by 2π. (Recall that momentum is mass times velocity.) As Heisenberg saw it, a high-precision measurement of either one of these two variables invariably "messes up" our knowledge of the other. Niels Bohr wasn't so happy with this "uncertainty" interpretation: see question 1.
(To be more specific than I was in class, it's actually the position and momentum along the same direction that can't both be known. For example, one can't simultaneously know an electron's x-coordinate and the x-component of its momentum [= mass times speed along the x-axis], and similarly for y and z -- but one can simultaneously know, say, the electron's x-coordinate and the y-component of its momentum to arbitrary precision.)
4.
One electron makes one bright flash at one spot on a phosphor screen, or makes one bright spot on developed film. There's no pattern involved -- just a single "blip." This is why we insist that electrons have particle properties: any time we look for one, we find it at a single place. One electron doesn't make a spread-out pattern.
This isn't the whole story, though; see the next question.
5.
Although each electron makes a single "blip" in a seemingly random place, when you keep track of many such electrons it turns out that these places aren't so random after all. Instead, they trace out a double-slit interference pattern! We never know just where the next electron will show up, but we can be sure that it will not be in one of the gaps of this pattern.
This is weird. Remember, a double-slit pattern is determined by the slit spacing -- that is, by both slits. So even though our electron goes through just one of the slits, the slit it doesn't pass through has some say in where it lands.
Should we conclude that this is impossible, and hence that electrons aren't particles? Waves, of course, would go through both slits at once. But then why does each electron make just one tiny blip on the screen or film, as in Question 4? Nature laughs haughtily at our confusion....
Note that photons are weird in precisely the same way, as can be seen by using a lamp so faint that only one photon at a time goes through a double slit.
6.
Quantum mechanics doesn't allow us to predict where a single electron will land, but it does allow us to compute some extremely valuable quantities: the probabilities that it will land in various places. If you don't think that probabilities represent valuable information, any casino operator can set you straight. The house can be quite sure of making money, regardless of occasional big payouts, so long as enough chumps -- er, players -- show up to cover rent and payroll.
Rather than computing the weekly take from one-armed bandits, physicists compute the probability that an electron will create a flash of light at a certain place on a screen, or that an alpha particle will be scattered through 90 degrees by a gold nucleus, or that a hydrogen atom at the third highest energy level will "jump" to the second level by emitting a red photon, etc.
7.
At what point during a measurement do submicroscopic objects like electrons "decide" what choices to make? The "many-worlds" interpretation answers this question by rendering it irrelevant: Nature makes all possible choices simultaneously! For example, an electron passed through a pair of slits causes the universe to "split" in some abstract sense. In each of these spin-off universes, a flash of light occurs at a different place on a phosphor screen. Every submicroscopic quantum "choice" is in fact a splitting of this kind, quickly resulting in a ridiculously large number of "parallel universes."
(A more concrete, albeit unethical, example: When we open the box to check on Schrödinger's cat, two universes branch off, one of which includes a living cat and one of which includes a feline corpse.)
Although I've oversimplified a bit -- it's actually the measurer who splits, not the entire universe -- the idea is still bizarre. Most physicists consider this "solution" to be even worse than the problem it's meant to solve, and hence reject it. I point it out here as an example of the weird stuff our data force us to consider.
8.
The "probability cloud" tells us where an electron is most likely to turn up, should we look for it. This doesn't mean that the electron was in that location all along. No, it was "smeared out" in an incomprehensible way, and the act of measurement caused it to "roll the dice" and show itself at one particular location. The electron cloud (a visual depiction of the electron's wave function) tells us the odds that this location or that will be the result of this dice-rolling exercise (the "collapse of the wave function").