Llama badges are a game on your account to collect as many llamas as possible. Your llama icon "levels up" and changes the more llamas you're given. Some people use it just as a game, others use it as a status symbol, and others as a way of showing their involvment here on DA. ^^
If you go to my user page, you see the button marked Give? It's below the llama icon. You click it, and then you click on the Give Llama Badge option which pops up on the menu. A window pops up, and you put your password into the space provided to authorize that you're giving the person a llama. Then you click okay or whatever, and off it goes. :3 Its pretty simple!