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.

Geeksn0w
1 Tethered jailbreak

p0sixspwn
1 Use redsn0w to tethered jailbreak and then upgrade to untethered by downloading the p0sixspwn package from Cydia.

purplera1n
1 Only applicable for devices with the old bootrom.