I decided to give quantum entanglement a page to itself because, to me, it is one of the most bizarre aspects of Quantum physics, which is itself a bizarre topic! Quantum entanglement was identified as an implication from the early days of quantum theory. Albert Einstein, Boris Podolsky, and Nathan Rosen proposed the EPR paradox in 1935 as an example of entanglement which Einstein referred to as "spooky action at a distance". It was not until 1980 that Alain Aspect was able to demonstrate the effect experimentally. Nothing in classical physics prepares anyone for the strange consequences of quantum physics, and entanglement is one of the strangest.
Consider the simple example of a decay process that produces a pair of particles, electrons for example, that must be identical except for having opposite values for their spin. This is a real example known as "spin anti-correlation", and must be observed to adhere to conservation laws. Quantum physics tells us that one particle has positive spin, while the other is negative, but not which is which. An essential aspect of Quantum Physics is that the original uncertainty about a particle's quantum state is a fundamental property of Quantum Theory. In this example, prior to making the measurement the particle does not have a definite spin state, but exists in a superposition of all possible spin states. It is not due to a lack of knowledge; it is an inherent characteristic.
So we let these two particles move further and further apart, and then measure the spin of one. This causes the "collapse of the wave function", as defined in the Copenhagen interpretation of quantum physics, and we now know the spin of the first particle. Instantaneously, however, we now know the spin of the other particle, which could be a very long way away, without having to measure it. This is in direct contravention of the Copenhagen interpretation, which states not only that we cannot know the spin of a particle until it is measured, but that the particle is in a state of having all possible spins until it is measured. It also implies that the "influence" travels faster than the speed of light; a direct contravention of the Special Theory of Relativity. In 2008, an experiment in Switzerland verified that the influence did indeed travel at speeds at least 10,000 times faster than light speed, though theory suggests that the effect is instantaneous.