When you have finished working on a ticket you should mark it "complete" to prevent it from appearing in the open ticket listings and indicate it no longer requires action. Your "complete" status can be called anything you like, all you need to do is specify which status is the "complete" in the setup. Go to Setup > Helpdesk > Helpdesk settings and select the complete status.

Mark a Ticket Complete

To mark a ticket complete change its status to your specified complete status. this can be done from within the ticket or update multiple ticket statuses using the batch processing.