Jailbreaking an iOS device is the process of taking away the restrictions Apple places on their devices giving the user root access, access to the raw file-system, and the ability to run software not signed by Apple.
Jailbreak refers to the process by which a user can enable the ability to run software that has not been explicitly approved by Apple. For an app to be run on an iPhone, it must first be digitally "signed" by Apple, so that the iPhone recognizes it as authorized. Jailbreaking allows users to run "unsigned" applications, by removing the restrictions that exist on the iPhone.
There is a large and active jailbreak community for the iPhone consisting of many users as well as developers. Cydia is the jailbreak equivalent of the App Store and serves as a central method of accessing software for jailbroken phones. It is usually installed by default by most of the popular jailbreak methods.
Jailbreak software includes a range of games, themes, and all other sorts of systems tweaks. Jailbreak software has historically given the iPhone features that Apple did not add until later, such as copy and paste, Skype over 3G, and wifi sync. Also, many applications that have been rejected by Apple will end up on Cydia, often because Apple does not approve of some aspect or functionality that the application provides to users, e.g. Google Voice applications before Apple changed their mind.
Jailbreak was previously deemed illegal under the DMCA until the Library of Congress issued a specific exemption for the situation. Apple will not support jailbroken iPhones, however, since it does violate their EULA. This is generally not a problem, since a jailbroken phone can almost always be restored to its plain, "vanilla" state.