Why is Nash equilibrium such an important solution concept? I'm taking an introductory course in game theory and we've recently reached the chapter that discusses Nash equilibrium. The concept itself is clear to me; what's not entirely clear, however, is the superlatives used to describe its importance. The book I'm using (by Solan, Maschler and Zamir) describes it as 

the most central solution concept for strategic form and extensive
  form games.

You can find similar statements in many more sources, often without proper justifications. 
The book does try to justify the importance of the concept by explaining the way it represents the notion of  stability. It claims that stability is a desirable property for any reasonable prediction about the game. Indeed, if we have a predicted situation that is not stable, one of the players will benifit from changing his strategy and the "prediction" will not come true. 
To put it differently, if all the players agree on some strategy profile, such a profile must be an equilibrium for the agreement to actually happen, without anyone wanting to break it.
That's perfectly true, but the thing I don't really get is: how will such an agreement ever take place in the first place? After all this is a non-cooperative game. Isn't it problematic that Nash requires every player to somehow "know" the strategy of the others while choosing his own strategy?
Maybe you'd like to say that each player is rational enough to figure out the equilibrium (assuming that it exists and unique) and then assume that the other players are going to follow it. But then I ask, why would they use that specific solution concept to predict the behavior of the other players? Is it because it is "the most important" one? Because that would be a silly circular argument. 
And even if we agree that Nash equilibrium is very important and useful, why is it widely agreed to somehow be "the best one"? Why is it better that the concept of eliminating dominated strategies, or the concept of maximin strategies, or any other solution concept?
As you can see, I'm new to the subject and have many more questions than answers. I hope you could help to clear things up a little bit more. 
 A: If for no other reason, Nash equilibrium and its refinements are the most important solution concept because they are by far the solution concept most commonly used, at least in the social sciences. 
The reason why Nash equilibrium is used so much could be accidental. People often attribute the following quote to Roger Myerson:

If there is intelligent life on other planets, in a majority of
  them, they would have discovered correlated equilibrium
  before Nash equilibrium.

You are right that it is difficult to justify the concept of Nash equilibrium as "the best" solution concept, especially in settings without communication. For a long time, people believed that Nash equilibrium could be deduced from the rationality of the players. It turns out that this is not the case (e.g., Brandenburger, 1992). People also hopped that behavior over time would converge to Nash equilibrium as the players acquire experience. There are some positive results for some specific settings, but not in general (see Nachbar, 2005). 
There is some empirical evidence that supports the use of equilibrium in some settings (e.g., Chiappori and Groselclose, 2002 or Walker and Wooders, 2001). But only in the context of zero-sum games. In such games, Nash equilibrium and minimax equilibrium coincide. 
The papers I references are very accessible, except for the Nachbar paper.  
