A 4-sided die is a pyramid. A 2-sided die is a coin. What's a 1-sided die? It's a Möbius strip. No matter how you throw it, the die will always land on its only side. Awesome Dice developed it for role-playing games, such as Dungeons & Dragons.
Now to the obvious question: why would you ever need a 1-sided die?
It's a way of resolving conflicts that come up in gaming sessions. Sometimes there are argumentative players who press a game master to randomly roll outcomes rather than verbally lead the players along a storyline. When you have such a player in your group, get out your 1-sided die.
The creators put it this way:
Imagine this all too common scenario faced every day by D&D DMs around the world: your group is in a dungeon crawl. There’s a room with a pie in it, and the pie is guarded by an orc. If the characters open the door, one of these things happen:
1. The orc attacks
That’s it, there’s only the one option. So the characters do indeed open the door and the poor DM consults the chart. Unfortunately the smallest die type he has is a d4, so our DM has no choice but to roll the d4 over and over until he gets a 1 before he knows what the orc is going to do.
This is exactly the situation that 1-sided dice are designed to solve. Now the DM can grab the d1 and quickly roll just one time, see what the orc will do, and get on with the action of the game. In this way 1-sided dice make your game move faster and better.
-via Marcel Ledbetter