Why was John Nash's 1950 Game Theory paper such a big deal? I'm trying to understand why John Nash's 1950 2-page paper that was published in PNAS was such a big deal. Unless I'm mistaken, the 1928 paper by John von Neumann demonstrated that all n-player non-cooperative and zero-sum games possess an equilibrium solution in terms of pure or mixed strategies. 
From what I understand, Nash used fixed point iteration to prove that non-zero-sum games would also have the analogous result. Why was this such a big deal in light of the earlier work by von Neumann? 
There are two references I provide that are good: One is this discussion on simple proofs of Nash's theorem and this one is a very well done (readable and accurate) survey of the history in PNAS. 
 A: As as been rightly said, Nash defined a concept of equilibrium for zero-sum games with $n$ players, and proved the existence (but no uniqueness of course) of such, while Von Neumann and Morgenstern did that only for $n=2$ (or larger $n$ but with very strong hypotheses on the game that reduces the problem to a game with $n=2$ players). But it is important to note than while doing so, Nash also defines a concept of equilibrium for non-zero sum games with $n$ players, for such a game is equivalent to a zero sum game with $n+1$ players: just add one new player, the "bank", whose gain/loss is defined as the negative of the sum of the gains of each other players. 
That being said, the real-world meaning of the concept of Nash equilibrium is very tricky, and it is far from clear if/when that concept is the right one to analyze a game situation, while in the case $n=2$, the Von Neumann/Morgenstern concept is much more obviously the only right one.
A: I think von Neumann dealt with the case $n=2$, and it was by no means obvious how to extend the concept of equilibrium for the general case and prove that it always exists. More precisely, $n$ players before Nash were reduced to the $n=2$ case by partioning the players into two groups in all possible ways. Once you regard several players as a single player, they are meant to cooperate as they must act like a single player. Nash is very clear about this in his 1951 Annals paper:

Von Neumann and Morgenstern have developed a very fruitful theory of 
  two-person zero-sum games in their book Theory of Games and Economic Behavior. 
  This book also contains a theory of $n$-person games of a type which 
  we would call cooperative. This theory is based on an analysis of the interrelationships 
  of the various coalitions which can be formed by the players of the 
  game. 
Our theory, in contradistinction, is based on the absence of coalitions in that 
  it is assumed that each participant acts independently, without collaboration or 
  communication with any of the others. 
The notion of an equilibrium point is the basic ingredient in our theory. This 
  notion yields a generalization of the concept of the solution of a two-person zero-sum game. 
  It turns out that the set of equilibrium points of a two-person zero-sum game is 
  simply the set of all pairs of opposing "good strategies." 
  In the immediately following sections we shall define equilibrium points and 
  prove that a finite non-cooperative game always has at least one equilibrium 
  point. We shall also introduce the notions of solvability and strong solvability 
  of a non-cooperative game and prove a theorem on the geometrical structure of 
  the set of equilibrium points of a solvable game. 

A: Let me start with my own view: 

The Nash equilibrium concept and the accompanying existence theorem is
  among the very few cornerstones in mathematical modeling of our
  reality .

Let me add a few more links. A paper by Ariel Rubinstein entitled "John Nash the master of economic modeling" and a document based on a seminar entitled: "The work of John Nash in Game theory." This includes the following quote from an earlier paper by Aumann. 
 
Nash equilibrium is probably the main reason for the "game-theoretic revolution" in theoretical economics.   
A: This answer overlaps with other answers but I think another restatement may be helpful because the situation is slightly confusing.
After the two-person zero-sum result, it is natural to ask about extending the results to $n>2$ and to non-zero-sum games.  Sometimes it is stated that Nash was the first to carry out this extension, but this is slightly misleading, because von Neumann and Morgenstern did consider both $n>2$ and non-zero-sum games and proved various things about them.  However, the key point is that it's important to ask the right question.  Intuitively, the basic question in game theory is to find the "optimal strategy", but it's not immediately clear what this means in an $n$-person non-cooperative game.  We now understand, thanks to Nash, that a basic necessary condition for a set of strategies to be "optimal" is for them to form a Nash equilibrium, but von Neumann and Morgenstern did not hit on this concept.  When they treated $n$-person games, they addressed different questions, such as what happens if the players form two coalitions.  So Nash didn't just answer the obvious question; the right question wasn't obvious, but he found it anyway, and answered it.
The second innovative aspect of Nash's work is that the two-person zero-sum result was based on the theory of linear programming and minimax.  Proving the existence of a Nash equilibrium requires different techniques.  So the naive approach to generalization, namely staring at the existing result and trying to figure out how to use the same ideas to prove something more general, does not lead to Nash's key insight.
A: The significance is best interpreted in conjunction with Nash's accompanying work.
Myerson gives a good history of the theory: Nash equilibrium and the history of economic theory
Here are some important points:

Thus von Neumann (1928) argued that virtually any competitive game can be modeled by
  a mathematical game with the following simple structure: There is a set of players, each player
  has a set of strategies, each player has a payoff function from the Cartesian product of these
  strategy sets into the real numbers, and each player must choose his strategy independently of the
  other players. ...
Von Neumann did not consistently apply this principle of strategic independence,
  however. In his analysis of games with more than two players, von Neumann (1928) assumed
  that players would not simply choose their strategies independently, but would coordinate their
  strategies in coalitions. Furthermore, by his emphasis on max-min values, von Neumann was
  implicitly assuming that any strategy choice for a player or coalition should be evaluated against
  the other players' rational response, as if the others could plan their response after observing this
  strategy choice. Before Nash, however, no one seems to have noticed that these assumptions
  were inconsistent with von Neumann's own argument for strategic independence of the players in
  the normal form.
Von Neumann (1928) also added two restrictions to his normal form that severely limited
  its claim to be a general model of social interaction for all the social sciences: He assumed that
  payoff is transferable, and that all games are zero-sum. 

In contrast, Nash provided a way to deal with the more general problem of non-transferable utility and non-zero-sum games.

But the most important new contribution of Nash (1951), fully as important as the general
  definition and the existence proof of Nash (1950b), was his argument that this noncooperative
  equilibrium concept, together with von Neumann's normal form, gives us a complete general
  methodology for analyzing all games....
  Von Neumann's
  normal form is our general model for all games, and Nash's equilibrium is our general solution
  concept. ...
Nash (1951) also noted that the assumption of transferable utility can be dropped
  without loss of generality, because possibilities for transfer can be put into the moves of the game
  itself, and he dropped the zero-sum restriction that von Neumann had imposed.



*

*Reference: Myerson, Roger B. "Nash equilibrium and the history of economic theory." Journal of Economic Literature 37, no. 3 (1999): 1067-1082.

