Game theory is the science of strategy in situations that involve more than one actor. This can include actual games, military battles, business interactions or managerial economics. According to game theory, the right strategy for an individual might be the same no matter how other players act. This is the dominant strategy. On the other hand, the Nash equilibrium doesn't describe a strategy as much as a stasis of understanding; each player understands the other player's optimal strategies and takes those into consideration when optimizing his own strategy.
Overview of the Nash Equilibrium
The Nash equilibrium is named after John Forbes Nash, who penned a one-page article in 1950 (and a follow-up in 1951) describing a stable-state equilibrium in a multiperson situation where no participant gains by a change in his strategy as long as the other participants also remain unchanged.
The most famous example of Nash equilibrium is the prisoner's dilemma. In the prisoner's dilemma, two criminals are captured and interrogated separately. Even though each would be best off by not cooperating with police, each expects the other criminal to confess and reach a plea deal. Thus, there is a conflict between group rationality and individual rationality, and each criminal is likely to rat the other out.
This example has caused some confusion about the Nash equilibrium. The theory is not used exclusively for situations where there is a defecting party; the Nash equilibrium can exist where all members of a group cooperate or where none do. In fact, many games can have multiple Nash equilibria.
Overview of Dominant Strategy Solution
It's possible that a dominant strategy solution is also in Nash equilibrium, although the underlying principles of a dominant strategy renders Nash analysis somewhat superfluous. In other words, the cost and benefit incentives don't change based on other actors.
In the dominant strategy, each player's best strategy is unaffected by the actions of other players. This renders the critical assumption of the Nash equilibrium—that each actor knows the optimal strategy of the other players—possible but almost pointless. (For related reading, see: Game Theory: Beyond the Basics.)