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 see a delete link in the bottom right of the menu. There is no way to get a deleted card back. It will not show up in any search results and actions associated with it will by removed from the activity stream.
You can also delete from the archive. Click "Archived Items" in the board's menu, find the card and select "Delete".