The Classic Quantum Entanglement Example
The classic example of quantum entanglement is called the EPR paradox. In a simplified version of this case, consider a particle with quantum spin 0 that decays into two new particles, Particle A and Particle B. Particle A and Particle B head off in opposite directions. However, the original particle had a quantum spin of 0. Each of the new particles has a quantum spin of 1/2, but because they have to add up to 0, one is +1/2 and one is -1/2.
This relationship means that the two particles are entangled. When you measure the spin of Particle A, that measurement has an impact on the possible results you could get when measuring the spin of Particle B. And this isn't just an interesting theoretical prediction, but has been verified experimentally through tests of Bell's Theorem.
One important thing to remember is that in quantum physics, the original uncertainty about the particle's quantum state isn't just a lack of knowledge. A fundamental property of quantum theory is that prior to the act of measurement, the particle really doesn't have a definite state, but is in a superposition of all possible states. This is best modeled by the classic quantum physics thought experiment, Schroedinger's Cat, where a quantum mechanics approach results in an unobserved cat that is both alive and dead simultaneously.
The Wavefunction of the Universe
One way of interpreting things is to consider the entire universe as one single wavefunction. In this representation, this "wavefunction of the universe" would contain a term that defines the quantum state of each and every particle. It is this approach that leaves open the door for claims that "everything is connected," which often gets manipulated (either intentionally or through honest confusion) to end up with things like the physics errors in The Secret.
Though this interpretation does mean that the quantum state of every particle in the universe affects the wavefunction of every other particle, it does so in a way that is only mathematical. There is really no sort of experiment which could ever - even in principle - discover the effect in one place showing up in another location.