Think about this: An interesting example given to illustrate two competitors who mistrust each other but still want to collude/compete at the same time. Criminals A and B are caught. Police tell A that if he confesses he will go free and B will get 5 years in the slammer. B is told the same thing. So the matrix is now: if A & B are silent both get 1 year. A confesses & B is silent then A goes free and B gets 5 years. A is silent and B confesses means A gets 5 years and B goes free. If both confess both get 3 years. If you are A what is your dominant strategy?
The traditional dominant strategy says that in order to reach maximizing equilibrium each party should take a position regardless of what the other party is doing or can do. So if A confesses he either goes free (if B remains silent) or gets 3 years(if B talks). So either way, whatever B does, A is better off or in the same situation as B.
But think about the situation where a dominant strategy is not possible. Or two "best situations" exist. Nash's equilibrium was illustrated with the example of the classic "battle of the sexes". Let us say a man and woman wanted to go out. The woman wants to go for a ballet and the man wants to go for prize-fighting show. They can only go to one show. The woman gets 1 unit of happiness if she goes to the ballet. The man gets 1 unit of happiness if he goes to the prize fight. They both get 1 unit of happiness if their partner accompanies them. They get -1 unit of happiness if their partner does not accompany them. Think about this situation as a 2 X 2 matrix with man as Y axis and woman as X axis. 0,0 represents man and woman together in prize-fight competition(man =2, woman =1). 0,1 represents man in prize fight and woman in ballet (man=-1, woman =-1). 1,0 represent man in ballet and woman in prizefight(both are so unhappy that man and woman each are -5 units unhappy). 1,1 is where man and woman are in ballet (man =1, woman =2). When man and woman sit and decide where they want to go, how do they decide? Their decision should result in a large amount of happiness units. Now clearly a more sophisticated dominant strategy is required.
Nash changed this to say that in order to acheive happiness/profit maximizing equilibrium A should do the best he/she can given what B does. B does his/her best given what A does. If you think A and B are like Coke and Pepsi fighting for market space, Nash's equilibrium changes the way firms thought about dominant strategy.