Jailbreak

Jailbreaking is the process by which full execute and write access is obtained on all the partitions of the Apple TV, iPad, iPad mini, iPhone or iPod touch. It is done by patching /private/etc/fstab to mount the System partition as 'read-write'. This is entirely different from an unlock. Jailbreaking is the first action that must be taken before things like unofficial activation (hacktivation), and unofficial unlocking can be applied.

The original jailbreak also included modifying the AFC service (used by iTunes to access the filesystem) to give full filesystem access from root. This was later updated to create a new service (AFC2) that allows access to the full filesystem.

Modern jailbreaks also include patching the kernel to get around code signing and other restrictions. These are called Kernel Patches.

Note: The legality of jailbreaking your device varies with each country/region. Wikipedia has a summary of legality for some countries.

Version numbers are the first to jailbreak and last is the last supported version. Last will only be listed if a newer version is out that does not support the device and iOS.

Note: The Apple TV 3G has never been jailbroken.

3.x
1 An earlier version of Cydia is installed.

5.x
1 Requires an unofficial PwnageTool bundle.

Apple TV
1 No package management GUI.

6.x
1 Tethered jailbreak on devices not vulnerable to 0x24000 Segment Overflow. 2 Point at 6.0 IPSW. Also Tethered jailbreak on devices not vulnerable to 0x24000 Segment Overflow. 3 Tethered jailbreak on devices not vulnerable to 0x24000 Segment Overflow. Requires pointing to 6.0 IPSW. Upgradable to untethered jailbreak via p0sixspwn Cydia package.

Apple TV
1 Tethered jailbreak. 2 Point at 6.0 IPSW. Also Tethered jailbreak.

7.x
1 Build 11A466. Requires a hex edit of evasi0n7 binary. See this. 2 Semi-Tethered jailbreak 3 Tethered jailbreak. Upgradable to Semi-Tethered jailbreak via repo http://geekrepo.beiphone.it package

Apple TV

 * There is no jailbreak available for the Apple TV 2G on 7.x.