All you need to do is head to the admin console, go to configuration > general configuration, and enter the status you want the ticket to go into when the user hits the reopen button.
This status needs to exist on your board or queue that you want to allow clients to reopen tickets on. If the status does not exist on a particular board, or this field is blank; the reopen button will not appear.
Note that your clients will only be able to reopen a ticket within 30 days of the closing date.
In Portal 2.0
In Portal 1.0
By clicking on this button the client will be prompted for a reason they are reopening a ticket. This will be left as a ticket note, and the ticket will be moved into the reopened status.