One of the biggest questions I hear all the time is people asking the difference between tethered and untethered. It’s important to not get the term confused with Internet Tethering which is sharing your WiFi from your mobile device to other devices. This type of tethering refers to connection through a cable with iTunes.
Your best option is untethered, let me explain.
Tethered means you are going to have to connect your phone with a cable into iTunes. This is not the best option because if you phone reboots or shuts off for whatever reason you will need your computer in order to start it or you will stare at the Apple logo on your phone and make it a very expensive screensaver phone. This also tends to be the first type of jailbreak that becomes available since it is the easiest to Jailbreak.
The untethered jailbreak option is by far the best. This does not require you to be plugged into iTunes. You can reboot your iPhone and it will simply start right up. Of course you might need to restart Springboard once in a while as this can be a common problem with your Jailbroken iPhone.
If you need some more understanding about the different parts of a Jailbroken iPhone check out the following links.