Every board has its own archive. The archive is the long-term holding area for tickets that have been completed or abandoned and aren't likely to be referenced any time soon. Moving tickets there after the work is done helps keep your board uncluttered. Archived tickets are still available in search and by viewing the board's archive.
To view a board's archive, visit the board, click the ellipsis menu in the upper right corner, and choose View archive.
To find a specific archived ticket, rather than opening the archive, you'll probably find it faster to use the search box in the navigation bar. It includes archived tickets.