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 (the menu 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".