Entanglement is "spooky action at a distance", as Einstein liked to say (he actually did not like it at all, but at some point he had to admit that it exists.) In quantum physics, two particles are entangled when a change in one particle is immediately associated with a change in the other particle. Here comes the spooky part: we can separate our "entangled buddies" as far as we can, they will still remain entangled. A change in one of them is instantly reflected in the other one, even though they are physically far apart (and I mean different countries!)
Entanglement feels like magic. It is really difficult to wrap our heads around it. And yet, entanglement is a real phenomenon, measurable and reproducible in the lab. And there is more. While for many years entanglement was thought to be a very delicate phenomenon, only observable in the infinitesimally small world of quantum physics ("oh good, our world is immune from that weird stuff") and quite volatile, recent evidence suggests that entanglement may be much more robust and even much more widespread than we initially thought. Photosynthesis may happen through entanglement, and recent brain data suggest that entanglement may play a role in coherent electrical activity of distant groups of neurons in the brain.
Entanglement is a good cognitive chunk because it challenges our cognitive intuitions. Our minds seem built to prefer relatively mechanic cause-and-effect stories as explanations of natural phenomena. And when we can't come up with one of those stories, then we tend to resort to irrational thinking, the kind of magic we feel when we think about entanglement. Entangled particles teach us that our beliefs of how the world works can seriously interfere with our understanding of it. But they also teach us that if we stick with the principles of good scientific practice, of observing, measuring, and then reproducing phenomena that we can frame in a theory (or that are predicted by a scientific theory), we can make sense of things. Even very weird things like entanglement.
Entanglement is also a good cognitive chunk because with its existence it implicitly whispers to us that seemingly self-evident cause-and-effect phenomena may not be cause-and-effect at all. The timetable of modern vaccination, probably the biggest accomplishment in modern medicine, coincides with the onset of symptoms of autism in children. This temporal correspondence may mislead us to think that the vaccination may have produced the symptoms, hence the condition of autism. At the same time, that temporal correspondence should make us suspicious of straightforward cause-and-effect associations, inviting us to have a second look, and to make controlled experiments to find out whether or not there is really a link between vaccines and autism. We now know there is no such link. Unfortunately though, this belief is very hard to eradicate and is producing in some parents the potentially disastrous decision of not vaccinating their children.
The story of entanglement is a great example of the capacity of the human mind of reaching out almost beyond itself. The key word here is "almost." Because we "got there", it is self-evident that we could "get there." But it didn't feel like it, did it? Until we managed to observe, measure, and reproduce that phenomenon predicted by quantum theory, it just felt a little "spooky." (It still feels a bit spooky, doesn't it?) Humans are naturally inclined to reject facts that do not fit their beliefs, and indeed when confronted with those facts, they tend to automatically reinforce their beliefs and brush those facts under the carpet. The beautiful story of entanglement reminds us that we can go "beyond ourselves," that we don't have to desperately cling to our beliefs, and that we can make sense of things. Even spooky ones.