Tethered jailbreak
A tethered jailbreak is only able to temporarily jailbreak the device during a single boot. If the user turns the device off and then boots it back up without the help of a jailbreak tool, the device will no longer be running a patched kernel, and it may get stuck in a partially started state, such as Recovery Mode. In order for the device to start completely and with a patched kernel, it must be "re-jailbroken" with a computer (using the "boot tethered" feature of a tool) each time it is turned on. All changes to the files on the device (such as installed package files or edited system files) will persist between reboots.
Using a tethered jailbreak
To boot tethered, their user must plug the device into a computer and run the jailbreak tool to "boot tethered".
If a user attempts to boot the device without using a computer, the device will either be stuck at the Apple logo, in Recovery Mode, or in a seemingly "un-jailbroken" state where Cydia, Mail, and Safari crash (and jailbreak-only tweaks/themes don't work) - until you boot tethered.
Tethered exploits
The limera1n Exploit and other bootrom exploits can be used to tethered jailbreak older devices on any iOS version, because bootrom exploits take advantage of code that is permanently embedded in the device's hardware, which Apple cannot update with iOS updates. Those tools do usually need minor software updates (not exploit-related) to explicitly support new iOS versions. They also use additional exploits (specific to each iOS version) to produce untethered jailbreaks when possible.
The initial jailbreak for the iPod touch (2nd generation) was tethered, until the hybrid dev team released the 0x24000 Segment Overflow. The codename for the tethered jailbreak was redsn0w Lite.
