A card can either be part of a publication, or unlisted, meaning that only people you share the URL with can read it.
Only members of a publication can create cards within that publication, but anyone can create an unlisted card.
Create a new card by going to the updates inbox, clicking "New Card", and selecting "Unlisted".
Create an unlisted card now, give it the title "My Test Card", enter "This is a silly card" in the first text chunk, and then click "Create Card" to create the card.
Next: Edit a Card