Tethered jailbreak

Tethered means being basically "attached" to your computer in a way; you must boot your device by running code on it (via the dock connector) to make use of an exploit.

iPod touch 2G

The initial jailbreak for the iPod touch 2G was tethered, until the hybrid dev team released the 0x24000 Segment Overflow. The codename for the tethered jailbreak was redsn0w Lite.

Linux

Currently, certain devices are capable of booting to Linux. The downside is that this is a tethered boot.

Explanation

Devices are tethered because there is one or more area where the device fails one or more signature check along the way due to the jailbreak. The device is able to boot up because there is a way to execute code via USB that allows you to bootstrap to a pwned (no signature check) iBSS, iBEC, or iBoot to finish the boot process.

redsn0w 0.9.6 beta 4

Because there is no exploit that allows the Kernel to be edited without it failing signature check every boot because the kernel does not check that the root filesystem is not edited, so you can instal Apps.