In discussing the idea of market failure, in class and in public talks, I usually start by pointing out that technical terms often sound self-explanatory and aren't; the theory of relativity, for instance, is not the theory that everything is relative. I go on to explain that "market failure" is an example of that problem. It does not describe all ways in which markets can fail and it is not limited to markets.
As I define it, market failure describes a situation where individual rationality does not lead to group rationality ..
Technical terms are tools for thinking and communicating. My definition of market failure makes it a better tool for those purposes than an alternative such as "a situation where the market does not allocate resources efficiently." The latter lumps together similar effects from unrelated causes—the market might produce the wrong allocation because the information to deduce the right allocation did not exist when the relevant decisions were made. It fails to identify situations where the same logic is functioning in the same way in different contexts—rational ignorance in voting as an example of the public good problem, prisoners betraying each other as an example of an inefficiently large output (of confessions) due to a negative externality.