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Here's a simple explanation why quantum entanglement is weird. Imagine that Alice tells you: "I have three numbered fair coins. I can flip one of them, but then the other two will disappear." Bob, who is far away from Alice and can't communicate with her, says: "I have another set of three coins that is mysteriously connected to Alice's and always gives the same outcomes."
1) If you ask Alice and Bob to flip the same numbered coin, their answers always agree.
2) If you ask Alice to flip coin 1 and Bob to flip coin 2, they are different about 5% of the time.
3) If you ask Alice to flip coin 2 and Bob to flip coin 3, they are different about 5% of the time.
4) If you ask Alice to flip coin 1 and Bob to flip coin 3, they are different about 20% of the time!
Why is that strange? Well, if coin 1 is different from coin 3, then at least one of them must be different from coin 2, which you didn't ask about, but could have. So the probability of the last situation can't be greater than the sum of the previous two. Even if Alice and Bob are lying to you and have agreed on some strategy beforehand, the above scenario is still impossible. But with quantum mechanics it's possible (see Wikipedia or Eliezer's post for details).
It's also easy to understand why such tricks don't allow you to send signals faster than light. For example, if you want to influence one of Bob's coins toward heads, asking Alice to flip one of her coins won't help you, because it's equally likely to come up either way. As a tool for playing games like this one, quantum entanglement is stronger than hidden variables, but weaker than communication.