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Jailbreaking, in terms of consumer electronics, is a practice of freeing a device from manufacturer-,or content creator-, enforced limitations over its use.
Essentially, to Jailbreak a device is the same thing as finding a software or hardware flaw that can be exploited to provide privilege escalation. Once achieved, a hacker can then access hardware or software functions that were previously off limits.
This can, for example, allow the execution of unauthorized / unsigned code, or it could be used to stifle digital rights management (DRM) technology so that a copy of an original item can be executed (or played).
The type of devices that are Jailbroken include smartphones (iPhone), games consoles (PSJailbreak for PS3, Reset Glitch hack for Xbox 360), handheld gaming devices (PlayStation Portable, R4 cartridges for Nintendo DS), consumer electronics devices (AppleTV etc.) and so forth. Basically, any consumer electronics device that has limitations on the software that can be run (or hardware than can be accessed or modified) can, in theory, be Jailbroken if a hack is found to overcome the limitations.
Generally speaking, once a hacker jailbreaks a device, they will release an easy-to-use hack for the device publicly, so that non-hackers who might not understand how or why the method works can take advantage of it.
The act of Jailbreaking is controversial, and whether or not you approve really depends on your circumstances and what angle you are viewing the act from. For a manufacturer or content creator, Jailbreaking may be a stepping stone to pirating games, or apps. In the case of games consoles, quite often the cost of assembling each unit is far higher than the cost of selling it to consumers. The intention is to get as many of the units on the market as possible, and make up the shortfall and extra profits from royalties through sales of third party software or peripherals (or directly with first party titles.) If then a device is Jailbroken to work in a way that has nothing with the intended purpose, it can be seen as a financial loss to the manufacturer.
On the other hand, Jailbreakers claim they should have a right to modify something they own in any way they see fit. The Electronic Frontier Foundation (EFF) advocates for the legalization of Jailbreaking in U.S. Copyright Law exemptions. The digital rights group has pointed out that many useful applications and bug fixes have been available for jailbroken devices but not for unaltered devices in the past, and that jailbreaking has proven to be part of innovation in the digital markets.