IANAS
So here's how I understand it (not being a scientist, but getting bought some variant of Quantum Theory for your Dog every year)
Say you've got two particles. You entangle them in such a way that a property called "magnetic north" (or "spin" - whatever) is opposite from each other*.
You leave each in a sealed box for ages & then you open one and measure it's magnetic north: It's pointing "up", so therefore, without opening the other box, you know the other particle's magnetic north is pointing "down"
There;s two ways of understanding that:
1) They were *already* pointing "up" and "down" before you put them in the box, so obviously if you find one is pointing up then the other must be pointing down - a bit like splitting a pair of gloves between two people, if one has a left then the other must have a right. This is "hidden values" and is classical.
2) Each particle is pointing both up & down *at the same time* in quantum superposition, and nobody knows anything about the up'ness or down'ness of either of them. When you open a box and measure one of them, the quantum superposition undergoes "wavefunction collapse"^ and turns out to be "up" classically. Somehow then, the other particle also undergoes wavefunction collapse, because you know it is pointing "down". Spookily. A bit like splitting a pair of gloves that are both left & right handed at the same time. Years later you look at your glove and find it is "right", magically the other person's (whose glove was until then both left & right) is now suddenly "left".
Clearly if 1 were true, nothing weird is happening. This Bell inequality thing (which is right over my head) is a way of telling whether 1 or 2 is true. Turns out that 2 is true.
* Up with respect to me.
^ yeah, I can use big words as well. You could also say that the wavefunction doesn't collapse, you're just in a different universe instead and the wavefunction is as it ever was.