Heisenberg's definition of his uncertainty principle ("The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa") may not be instantly clear, but here's a simple example that somehow makes it all remarkably easy to grasp.
In response to a Kinja-discussion on the explanations that finally made us really get scientific principles, commenter Vine shared this helpful example that helped them really get one of the slipperier concepts in theoretical physics:
I think it was my college optics course when I finally really understood the uncertainty principle (the position/momentum version, not The Measurement Problem). The professor was explaining how photons are wave packets, and that there was a fundamental uncertainty about their position and frequency. You can translate momentum directly into the frequency domain, so it's really the same problem. The explanation goes something like this:
Imagine that you have a perfect sin wave, to determine it's frequency, just measure the distance between two wave peaks of equal amplitude and invert. But where is the sin wave? To be a perfect sin wave, it has to be of infinite length, so in a very real way, it doesn't have a position.
Now imagine a Dirac delta function (a function of infinite amplitude at a single point), it has an extremely precisely defined position, but how do we measure the frequency? It has only one peak, so there's no 'between' to measure. Therefore, in a real way, it has no frequency.
So now consider the intermediate case, a wave packet has a frequency, but the peaks are at different amplitudes, so there's ambiguity. Do we measure the peaks at the center of the packet? From end to end and average? There's no correct answer, and the difference between those possible answers is the error measure, or uncertainty, in the frequency. Similarly, where is the packet? is it at the central peak? Does it have length from end to end? that distance is also uncertainty.
And now it's perfectly obvious why there is (and must be) a trade off in uncertainty between position and frequency. As the packet gets broader the frequency becomes more well define, but the position becomes more ambiguous. As the packet gets narrower, the definition of position becomes sharper, but the frequency becomes more ambiguous. Translated back into a momentum domain, it's easy to see why uncertainty is a fundamental property of physics.
Image: Werner Heisenberg lecturing via Lindau Nobel meeting