In quantum mechanics _everything_ about the position and velocity of a particle is described by a complex-valued function of its position, wavefunction psi(x). conj(psi(x))*psi(x), the square of the absolute value of psi, describes the probability density of finding the particle around the place x. Similarly, the probability density of finding the particle with a momentum around p is given by conj(psif(p))*psif(p), where psif(p) is the
Fourier transform of psi(x). So, a particle with a precisely defined position would have a wave function that
looks like a 'spike', which is zero everywhere except where the particle is. This kind of a function is called the "Dirac delta function" in math jargon. On the other hand, a particle with a precisely defined momentum would have a wave function that looks like a sine wave (The Fourier transform must be a Dirac delta function). So the uncertainty principle follows from the fact that the sine function and Dirac delta function are not the same function. It is impossible to have a function that has a precisely defined frequency, but is only non-zero at a single point.
In fact, one way to look at wavelets is that they are kind
of 'compromises' in this respect, being moderately localized in both time and frequency. This is part of what makes them so useful in representing different kinds of data.
Curiously, it turns out that the bell curve that describes the normal distribution is the best compromise between localization in space and frequency! This function as a wave function gives the equality in the uncertainty principle inequality.