Full Channel: The program and files are within the channel, so you don't need an SD card. However, that means you cannot easily update it. You would have to create and install a new channel every time you wanted to update. They are also much larger than forwarders.
Forwarder Channel: It is basically like a shortcut on your computer's desktop. The program is not part of the channel. All it does is search for the boot.dol in the SD card (and usually a USB drive), then launches it. Whenever you want to update, all you have to do is replace the old boot.dol in your SD card with the new one. They are much smaller than full channels.
There is also something called a "Hybrid Channel" which is a forwarder and full channel combined, but they are very large, and a forwarder should be enough.
The NeoGamma channel that I linked to searches here for the boot.dol: