Imagine a point. What is the wavelength of this point? If you're confused by this, you should be: a point is not a wave, so asking about its "wavelength" is silly.

Imagine a wave -- an infinitely long sine curve with crests and troughs that repeat forever. Where is the wave? This is a meaningless question -- it is equally everywhere at once, and so nowhere in particular.

If something is a point, it can have a position, but not a wavelength. If something is a wave, it can have a wavelength (the distance between crests), but not a position. Thus, in this sense no object can ever have both a position and a wavelength, since to do so it would have to be two completely different objects at the same time.

None of this is particularly deep. What is deep, however, is that it turns out that momentum and wavelength are physically equivalent quantities -- knowing one is equivalent to knowing the other! In fact, in a sense our everyday notion of momentum simply doesn't exist, but is just an illusion created by the true thing, the property of wavelength.

This is a very strange idea to swallow, and the only reason that we do is simply that we have done lots and lots of experiments and they all point towards it being true. Once you have accepted it, however, the uncertainty principle becomes easy to see: something cannot have both a precisely defined position and a precisely defined momentum at the same time because it cannot be simultaneously both a point and a wave.

The equations presented in the other write-ups of this node just state this idea in more mathematical terms, with one additional insight: it is possible for something to have both a rough position and wavelength by existing in a state that is neither a point nor a wave but something in between -- a shape that is spread out over space and yet concentrated in a particular area. The mathematical relation basically gives you a trade-off between how point-like and how wave-like something can be at the same time, which translates into a relation for the precision with which we can simultaneously measure position and momentum.