Ron Jeffries and his co-authors created a great summary of the user story process. The 3 Cs stand for Card, Conversation, and Confirmation. Going through those three steps in this order will enable you to deliver solutions to the users' problems.
The size, the shape, and what's on the back of the card are all irrelevant. As long as it's writable and movable, almost any object can serve as a medium for your story. With that said, a stone tablet is a bit less practical than a sticky note (often called Post-it note after the famous brand), so most people prefer sticky notes for story cards.
The biggest advantage is that they stay on a vertical surface, like a wall. Also, user story maps are often created on walls. This has many benefits. First of all, it forces people to stand up. This creates much better discussion dynamics, while also shortening the average length of the meeting, forcing people to focus, and optimizing the communication....