@mojosodope, correct to some extent. The portion about not booting tethered will result in jailbroken related apps/tweaks not running is slightly off. What you're trying to explain is a semi-tethered jailbreak. So here's where we blow your mind SAMMIE.
Tethered Jailbreak: Requires a PC when rebooting iPhone (respringing doesn't count). If you don't boot tether, you'll be walled with a Recovery Mode screen until you boot tether. No access to the phone's springboard whatsoever.
Semi-tethered Jailbreak: Requires a PC to get jailbroken apps working. With this type of jailbreak, you will have access to the springboard, which is what mojosodope is trying to explain. You can use stock apps like Phone, Message, Notes, Contacts, etc. But, your iPhone cannot use/start jailbroken apps/tweaks. It's kind of like safe mode in a way.
Untethered Jailbreak: Does not require a PC for anything except restoring if you want to wipe the jailbreak. You can reboot as many times as you like, whenever you like, and you won't need a PC to make stuff work again.
Untethered is the preferred jailbreak so users don't have to worry about needing a computer access to get their phones running again. But yes, if there is an untethered jailbreak that's released after a tethered jailbreak, in most cases, there will be a Cydia tweak that can make a tethered or semi-tethered jailbreak become an untethered by installing a tweak released to Cydia. One example is the Rocky Raccoon untethered exploit for iOS 5.1.1