In short, it's BB code.
For example, my signature looks like this:
[IMG]http://i211.photobucket.com/albums/bb260/Jan-san144/Touhou/A-FSig.png[/img]
You find the image you want (within the image and file size rules, of course!), upload it to the image hosting site of your choice (I like Imgur, but a friend recently introduced me to puu.sh), and stick it into the [img] tags, as above.