Let's say, for example, you have two quarters in your hands. Same mass, same feel, same everything, except one of them has a blue dot painted on it. If you close your hands and shake them around for a few minutes and then stop, letting each coin fall in a separate hand, then you have no idea which hand the blue dot coin is. These two coins now have a sort of entanglement, i.e. identifying a property about one coin (blue dot or not?) immediately tells you something about the second coin. If open only your left hand and find the coin with the blue dot, you know that the coin in your right hand does not have a blue dot (and vice versa).
Similar situations arise in quantum mechanics as well. For example, a pion is a neutral particle with a quantum spin of zero. Pions can decay into an electron and a beta particle (and energy). A beta particle is the antimatter equivalent of an electron, so that means all of its properties are the same except it has a negative charge.
So, if you have two particle detectors in the room and you notice that the beta particle and the electron strike different detectors, once you identify what Detector 1 detected (+ or -?) you immediately know what Detector 2 detected. This is because their quantum states are entangled. But not only can you learn what the second particle is. If you measure the first particle's spin (it will be +/- 1/2 because electrons are spin 1/2 particles), you immediately know the second particle's spin, since the total spin has to add up to zero (quantum spin is a conserved). These particles' spin is entangled as a part of their quantum states being entangled.