An exploit or security vulnerability is a programming oversight which allows us to break Wii Security and gain more access into the system. They are generally good for homebrew.
Exploits come in multiple flavors: IOS exploits such as the Signing bug are security vulnerabilities in the Wii's internal firmware, gamesave exploits such as Twilight Hack are security vulnerabilities in the savedata of games and channels, and System menu exploits such as Bannerbomb are security vulnerabilities in the way the System Menu handles data.
Gamesave and system menu exploits fall under the category of homebrew exploits which are generally user-applied in order to execute unsigned code. IOS exploits are used by homebrew code in order to grant further access to the system.
It is notable that most of these exploits are buffer overflows. One is a null pointer bug and one is a genuine confusion between string.h functions. Gamesave exploits tend to be the most numerous, since most save loading code is written in C with little attention paid to sane string handling. With one exception, Nintendo can only fix these exploits with a System Menu update.