Well there's probably nothing else that is so famous and yet causes so much confusion. The HCP lies at the heart of
Quantum Mechanics (as
Feynman puts it, "it protects Quantum Mechanics") and somehow it seems to have caught a lot of popular attention outside the
physics community also.
Well lets get the Mathematics out of the way first. So here's the generalized uncertainty principle:
If A and B are any observables (Hermitian operators) define
dA=A-E(A) , E(A) means expectation of A
dB=B-E(B)
then
E(dA2)E(dB2) >= (1/4)*|E(A,B)|2
When A=x and B = p, this reduces to the normal del(x).del(p) >= h/4pi because [x,p] = ih/(2*pi) .
Sakurai's book Modern Quantum Mechanics has a proof in the first chapter.
Okay now lets take the WU's one by one. Socialist Wolf's WU first. Well it is possible to determine the exact position and velocity that the particle had at a point in the past. So you can determine the position and then later determine the velocity and then say "look at t = -10 seconds the particle was here and had this velocity". Feynman explains this nicely in the first chapter of the third volume. One way to look at this is "Everything in the past is a particle, everything in the future is a wave". Thats not what the uncertainty principle deals with though.
Brazil's WU then. Here's the argument Heisenberg used first. Lets say you wish to measure the position of a small object accurately. So you must use light of a small wavelength to resolve the object. The smaller the wavelength, the larger the energy of the photon, so the larger the 'kick' imparted to the object. Thus the object will necessarily get disturbed if you try and measure its position accurately.
The problem with such an argument is that they assume that there is some true position of the particle out there. This is what Heisenberg thought initially though he was later convinced by Bohr.
What this means is that it is meaningless to speak of a dynamical characteristic in the abscence of measurement. Dynamical characteristics exhibit themselves only as a result of measurement. The first chapter of Landau and Lifshitz deals with this if anyone's interested.
Thus the correct way to look at this is to say that if you make a measurement of position the result of this measurement is necessarily probabilistic. You could find the particle here or you could find it there. The fuzziness involved (The standard deviation of the underlying probability distribution) is the uncertainity.
Finally Bitter_Engineer's idea about a photograph is an interesting way of showing how you cannot measure position and velocity with a single measurement but the uncertainity principle goes deeper than that. It has to do wth the fuzzy probabilistic nature of Quantum Mechanics itself.