Entanglement is a relationship between two or more systems such that observations of one provides information about another.
Perhaps the first thing one needs to know about quantum entanglement is how it compares to the classical entanglement that we are all familiar with (and which no-one gets particularly excited about). So let’s start with the classical case.
If we put two distinct items (eg black&white balls, left&right gloves, or Bertelmann’s socks) in identical boxes, and send them far apart after making a random choice to decide which box goes in which direction, then an observer who opens a box and sees one of the items knows immediately what’s in the other.
Before the observation, the state of the system is not known precisely but instead can be described as a probabilistic mixture which “collapses” as soon as the observation is made. But of course we all understand that the probabilistic nature of the situation depends on the observer, and the collapse seen by the first observer doesn’t happen for the second until either the second box is opened or a message from the first observer travels between them.
In the classical case, all of the observer-dependent randomness can be attributed to a lack of knowledge, but in the quantum situation at least some of the randomness appears to be essential in that sometimes we cannot determine one quantity without introducing randomness in another. It’s as if the boxes contained a pair of socks such that if one looks red then the other looks green, but if we look in different light the possible colours are yellow and blue. But a sock that looks either yellow or blue is equally likely in the original light to appear red or green.
Actually by cleverly packing random pairs of socks it is possible to reproduce those (red-green vs yellow-blue) quantum predictions in a perfectly classical way, but there are also intermediate situations, analogous to lightings in which the colours are orange and turquoise, and what quantum theory predicts about how an observation of either orange or turquoise affects the probabilities of red vs green and yellow vs blue is slightly outside the range of anything that could have been caused by any process of selecting from a random selection of pre-packed sock pairs.