Jailbreak: Getting root access to the device by “escalating user Privilege”, Privilege escalation is the act of exploiting a bug, design flaw or configuration oversight in an operating system or software application to gain elevated access to resources that are normally protected from an application or user. The result is that an application with more privileges than intended by the application developer or system administrator can perform unauthorized actions.
In iOS the user only have access to an specific area (commonly know as “Userland”) by the mean of a CHROOT jail, breaking out of this limitation allow regular user to scale to “root” (yielding the name Jail Breaking).
Tethered Jailbreak: is when a computer or another device (external Harddrive, modChip. etc) is required to inject the Jailbreaking code at boot. So once your iDevice is successfully jailbroken, you'll need to connect your iDevice to your computer if you ever need to reboot your iPhone or your iPhone loses battery charge.
Untethered Jailbreak: Means that the Jailbraking code live somewhere inside the iDevice and it can be invoke at boot without the need of any external device.