A Tweet ID is the unique long number that appears in the URL when you have clicked through to an individual Tweet. For example, when I posted a link on Friday to test this, it generated this Tweet with ID 863438724482334720 which clearly shows a card.
Sometimes, for a brand new page or article, our Twitter crawler may take a short time (or even a few hours) to generate the card preview. You can usually use the validator to check that your URL should perform correctly.