Two areas that have been profoundly affected by game theoretic research stemming from Nash's contribution are
There are actually a few examples of what would come to be known as Nash equilibrium in the industrial organization literature that predate Nash's work (for example, Cournot's 1838 analysis of oligopoly competition). However, until Nash (and Selten, Harsanyi, and others) made game theory a general purpose tool, industrial economics was primarily focused on relatively naive models of competition. In the last 30-40 years there has been a revolution in industrial organisation as economists have used game theory to essentially reinvent the study of market competition around oligopoly theory and the study of strategic interaction. Our modern understanding of consumer search, limit pricing, strategic entry and entry deterrance, predatory pricing, strategic advertising, switching costs, product differentiation, platform competition, horizontal and vertical integration, etc. are all predicated on models that rely mostly on Nash equilibrium (or a refinement thereof) as the solution concept. Jean Tirole was recently awarded the Nobel prize largely for work in this area.
This work has also found great practical application in areas such as antitrust policy. Prior to the 1960s, antitrust enforcement in the US (and, to a large extent, elsewhere) was inconsistent and based on unsound economic principles. A combination of the insistence by scholars (especially those based in Chicago) on more careful analysis, and the new tools of oligopoly theory have lead to a much more robust and well-grounded approach to regulating competition.
The study of auctions is game theoretic by its very nature: most auctions involve very direct strategic interaction between a relatively small number of bidders. It should come as little surprise, then, that auction theory essentially did not exist prior to the work of Nash (the formal study of auctions can be traced to W. Vickrey (1961) "Counterspeculation, Auctions, and Competitive Sealed Tenders," Journal of Finance 16(1); also the recipient of a Nobel prize).
None of the cornerstones of auction theory (revenue equivalence, the linkage principle, optimal auctions—source of yet another Nobel prize, etc.) would exist without the solution apparatus that can be traced to Nash. This work, too, has been of great practical importance. From radio spectrum licenses to carbon emissions permits, and from public procurement to Google ad auctions, auction theory has had a significant effect on informing good auction design. See Klemperer (2004) Auctions: Theory and Practice, Princeton University Press for an accessible summary of the theory and its applications.