The following video shows you how to enable the close button on a ticket
Autotask users don't have to configure anything - the ticket will be put into the Closed status without any setup.
Please note that this only works if you are running on SOAP API. Once you update to RESTful API, this set up in no longer valid.
If you are on RESTful, you can set the close status you want by setting the field in your general configuration tab in the admin console.