What does it do? I press it and nothing happens.

if you're on Windows, dnload and run icebox.... then log into game.

once in game, you get many different capabilities:

1. if you click on the dice, you'll see a popup that shows the raffle page
2. if you click a group in the find-premade list, you'll see the group leader's armory
3. anyone you target will also pop up their armory
4. clicking the eyes on the character pane would pop up your AskMrRobot page
5. clicking the wowhead icon on the quest description will pop up the wowhead page for the quest
6. if you click any of the link on the info panel, it'll jump to that page
7. if you use the icebox addon panel, you can shift click items to be looked up on wowhead or undermine journal
8. if you select skype in the icebox addon panel and type in the skype address, skype will dial it when you hit enter

etc

it's all dependent on icebox running in the background

Is their anyway to make the skype box not show up everytime you login?

the next version of the icebox addon will be hidden by default