Let's suppose that person A and person B are sitting together in a room when an angel appears in a burst of light and says "Be not afraid! Here are 100 grapes for your enjoyment - divide them fairly between the two of you, and then consume them." The angel vanishes.
How should A and B divide the grapes? As it happens, person A likes grapes twice as much as person B does. (For the purists, assume that each of them has constant marginal utility and that these facts are commonly known.) The following dialogue ensues:
A: In order to divide them fairly, we should split the grapes evenly - 50 to you, 50 to me.
B: That's not a fair division: I like grapes half as much as you, so if we split 50-50, I only end up half as happy as you are. A truly fair division would make us equally happy. Therefore, I should get 66 grapes, and you should get 34.
A: That's ridiculous! It can't be fair to give fewer grapes to the person who likes them more. Our individual happiness is irrelevant - fairness means splitting the grapes evenly, nothing more.
B: How can our individual happiness be irrelevant? The whole point of having the grapes is to makes ourselves happier. It's the quantity of grapes that is irrelevant here - neither of us cares about how many grapes we get except through how much utility we will gain from eating them. Our happiness is the relevant concept to equalize here.
A: [turns to a commentator] What should we do?
C: The fair thing to do is show equal concern for the interests of each of you. And B is quite correct that individual utility is what matters.
B: I knew it!
C: And since A would receive so much more utility from each grape, it would be unfair to give any at all to B. That would be to treat a small increment to B's welfare as more important than a larger increment to A's welfare. Clearly that is mistaken. You ought to give all 100 grapes to A.