A tethered jailbreak requires the iOS device to be plugged into a computer when turned on. If the iPhone is not plugged into a computer and booted with special software the device will not enter a jailbroken state.
Untethered jailbreaks do not require a computer. Everything required to enter the jailbroken state is contained on the iPhone. Rebooting away from a computer does not affect the jailbreak. iOS 4.3.3 and older jailbreaks including PwnageTool, limera1n, greenpois0n and sn0wbreeze are untethered solutions.