These are cards which, when played, force or allow you to trash one or more of your cards. Trashing means placing the card on the Trash Card, or Trash for short. Since no card (as of yet) allows retrieval from the Trash, trashing results in a card's permanent removal from the game.
From a beginner's perspective, it may seem that trashing one's own cards is counterproductive. See Why Trashing Is Good to understand why it can be a useful tactic.