Fair share is a piece that the person who received it believes it's worth 1/N (N is the number of the object's shares). Each person is assumed to act rationally with the intent of getting a fair share, while everyone has the ability to decide whether a share is fair or not. However, it is certain that the divider is going to divide an object or a set into equal parts, as he does not know which one he will end up with.
Let's say that there are three people trying to divide a cake. The lone divider cuts the cake in equal parts and the other two people (the "choosers") get to determine which share is fair for them. If they choose different parts, the division is simple: each one gets the share he chose, while the divider ends up with the remaining part. Since for the divider every share is fair, everyone is satisfied.
If the choosers believe that the fair share for them is the same part of the cake, then there is an additional step to the problem. On this occasion, the divider takes one of the unwanted pieces, while the remaining pieces (the piece both choosers want and the unwanted parts) are theoretically merged. From then on, the two choosers have to follow the divider-chooser method.
One person is randomly assigned to be the divider. A flip of a coin can determine that. Then, he gets to divide the remaining parts of the cake in two fair shares. However, it's the second person, the chooser, who decides which piece each one gets. With this in mind, the divider must cut two equal halves, as a larger half is bound to end to the chooser. Therefore, for the divider both shares are fair. The chooser can pick whichever share he finds fair and so both participants end up satisfied.