Entropy is expected information. That is, given a random variable, if you compute the expected value (the sum of the values weighted by their probability) of the information of an event (the log base 2 of the multiplicative inverse of the probability of the event), you get the formula for entropy.
Here it is explained at length: "An Intuitive Explanation of the Information Entropy of a Random Variable, Or: How to Play Twenty Questions": http://danielwilkerson.com/entropy.html
Here it is explained at length: "An Intuitive Explanation of the Information Entropy of a Random Variable, Or: How to Play Twenty Questions": http://danielwilkerson.com/entropy.html