Roughly, if two players are trying to divide up a surplus, how much should they each get? Is there some rational way to do so? Nash's answer is a mathematical model which says that if you assume:
* Pareto optimality (the solution is one of the best possible for the combination of the two players)
* Scale-independence (the result should be the same even if one person calculated in Euros and the other in dollars)
* Independence of irrelevant alternatives, and
then the resulting split *must* be the point at which the product (x1-d1)α(x2-d2)β is maximized, where x1,x2 represent the agreement; and d1,d2 represent the bargainers' "default" position of no bargaining. (You also need the agreement to be better than each party's next-best alternative.)
But what are alpha and beta? Well... that's where the problems start. They are the "bargaining power" of the two parties. Not their ability--- that is assumed to be equal--- but who has the advantage in the structure of the bargaining, due to risk aversion, or impatience, or some other factor. There is little way to fill in the values, other than the special case in which they are assumed to be equal. Which is worth studying, I suppose.
If you don't make that assumption, you can eliminate one of the variables but are left with a model which says that any amount within the given bounds can be reached depending on who has the upper hand in bargaining. Modern economics: making the trivially simple sound complicated for more than half a century so far! :)