An untethered jailbreak has the property that if the user turns the device off and back on, the device will start up completely, and the kernel will be patched without the help of a computer – in other words, it will be jailbroken after each reboot.
[crappy] With a tethered jailbreak, if the device starts back up on its own, it will no longer have a patched kernel, and it may get stuck in a partially started state; in order for it to start completely and with a patched kernel, it essentially must be "re-jailbroken" with a computer (using the "boot tethered" feature of a jailbreaking tool) each time it is turned on.
[you] A device with a tethered jailbreak may be able to have a semi-tethered solution, which means that when the device starts up on its own, it will no longer have a patched kernel (so it will not be able to run modified code), but it will still be usable for normal functions. With a semi-tethered solution, the user can also choose to start the device with the help of the jailbreaking tool in order for it to start with a patched kernel (jailbroken).
As @kobaltz mentioned, it is generally best to wait for an Untethered solution unless you have some major reason to require a jailbreak immediately. By jailbreaking, you also should wait on any updates of the system software since Apple actively works to inhibit jailbreaking (and an update will remove it 99.9% of the time).
What you should do --> Boot attached to your computer, there should be an option to boot tethered. (that is assuming you care for running jailbreak tweaks).