A tethered jailbreak is a jailbreak

A tethered jailbreak is a jailbreak where the device requires a computer to assist booting up. If the device's battery dies, or you turn off the device, or you reboot it, you will either be (A) stuck at the Apple logo or (B) boot up into a seemingly "un-jailbroken" state where Cydia, Mail, and Safari don't work - until you plug the device into a computer, open your tethered boot program (for example redsn0w), and follow its instructions. The situation in (B) is often called a semi-tethered jailbreak.

(Note: due to this requirement of booting your device with a computer, tethered jailbreaks are somewhat inconvenient. In general, people rarely need to reboot their devices unless they run out of battery, but a few popular Cydia packages also require a reboot after installing, including Winterboard and iFile. Most packages just require "restarting SpringBoard", which is not a reboot. So a lot of people don't mind having a tethered jailbreak, but you have to decide if this kind of jailbreak is OK with you.)

To boot tethered: if you're using redsn0w, plug in your device, open up redsn0w, click "Extras", and then click "Just boot". If you used a different tool to jailbreak, that tool probably also has an option to boot tethered.

If you're having problems getting your device to be recognized by redsn0w, start this process in DFU mode. To enter DFU mode: with your device connected to your computer, hold the power (lock) button and home button for 10 seconds, then release the power button but keep holding the home button for another 15 seconds.

If redsn0w still isn't booting your device properly, follow this guide to using redsn0w - including setting compatibility modes, running it as administrator, and specifying the IPSW. These steps solve many redsn0w problems.

Do I have a semi-tethered jailbreak?
Depending on the packages installed on a device with a tethered jailbreak, you may actually have a "semi-tethered jailbreak", which means that when you reboot your device without plugging it in, your device boots up seemingly "un-jailbroken" and Cydia, Mail, and Safari crash when you tap them. To fix that situation, boot tethered.

In more detail: tethered jailbreaks behave semi-tethered by default. If you install Cydia Substrate (Mobile Substrate) tweaks, your device will still be semi-tethered. But if you install Notification Center plugins that don't depend on WeeLoader, your device will no longer be semi-tethered - unless you also install the BigBoss semitether package.