Archive cards by clicking "Archive Card" in the Actions menu on a card's back. Or you can also select "Archive" from the card menu (click the pencil icon that appears on hover). You can also archive with the "c" keyboard shortcut when hovering your mouse over a card.
Click "Menu" then "Archived Items" from the board sidebar, find the card and click "Send to Board".
You can delete a card from the back of the card. Click the card to open it, then choose "Share and more" from the bottom right. You'll see a delete link in the bottom right of the menu.
You can also delete from the archive. Click "Archived Items" in the board's menu, find the card and select "Delete".