This writeup is a nonmathematical discussion of the Heisenberg Uncertainty Principle. The Heisenberg Uncertainty Principle is a mathematical inequality, but its philosophical interpretation is a fascinating and surprising aspect of our universe.
Some background information about quantum mechanics
Before the age of quantum mechanics, the goal of physics was to be able to deterministically predict the future universe given complete knowledge about the present universe. More practically, given the state of an isolated system in the universe, we should be able to predict how that system evolves with time. For example, if we want to predict where a bullet will land after we fire it, we shouldn't need to account for the gravitational pull of Jupiter. However, after accounting for Earth's gravity, air resistance, the curvature of the Earth, etc., we should be able to precisely determine where the bullet will land. Until the 20th century, physicists were very successful at developing equations that deterministically predicted the future.
In the early 20th century, physicists were puzzled by experiments that suggested that at the atomic/subatomic scale, deterministic predictions were impossible. Early such experiments were the Stern-Gerlach experiment and the double-slit experiment. Although determinstic predictions weren't possible, physicists found that probabilistic predictions were. Let's take the double-slit experiment as an example. Although it was not possible to know where an electron would be measured to land on a screen after going through two tiny slits, the probability of it being measured to land at a certain location could be predicted.
Physicists such as Pauli, Dirac, Schrodinger, and Heisenberg developed the mathematics of this probabilistic physics, and it became known as quantum mechanics. Many physicists (such as Albert Einstein) did not believe that the universe was truly probabilistic--they felt that we just didn't know everything there was to know. Experiments later in the century seem to prove that the universe really is probabilistic. Almost all physicists now accept the probabilistic nature of the universe as a fact.
While in general we cannot give the precise location, velocity, etc. of a particle, we can write functions that give the probabilities of measuring certain values for those "observables." A very common and important example, called the wavefunction and denoted by Ψ(r), describes the probability of measuring a particle to be at any location r in space. This function could be spread out over all space (suggesting that the particle is completely delocalized) or it could be concentrated in a tiny area (suggesting the particle is quite localized). Since the function describes probability (more exactly, probability density), all we require is that its integral over all space be 1, meaning there's a 100% chance that we'll measure the particle to be somewhere. We could also write a function Φ(v) that tells us the probability of measuring a particle to have some velocity v.
The Heisenberg Uncertainty Principle
Finally we can discuss the Heisenberg Uncertainty Principle. Consider the function Ψ(r). We can find the average value of Ψ(r), (i.e. the average position measured for the particle if we could measure its position several times). Furthermore, we can find a value called the uncertainty that describes the average difference in magnitude between the result of an actual measurement and the average measurement result*. Similarly, we can find the uncertainty of Φ(v).
* Uncertainty is actually defined as the root mean square of the difference, since the absolute value function is difficult to work with.
The Heisenberg Uncertainty Principle states that the product of the uncertainty of Ψ and the uncertainty of Φ must be larger than a nonzero, universal constant (Planck's constant/4π*mass). Philosophically, this means that a particle can NEVER have a precise location and velocity at the same time! Certainly this is a surprising result to everybody, but is it any more weird than the simple fact that we have to resort to probability functions to describe the position of a particle?
In reality, the Heisenberg Uncertainty Principle is a more general mathematical inequality that governs the product of uncertainties for all pairs of physical observables. Other writeups in this node go into more detail about this.