There is no Customer Support Agent in Town who does not look at the priority when handling a ticket. But yeah !!.... the kind of ticket also plays an important role. For eg : A ticket which says Password reset is to be given lesser priority than a Server down issue.
So the basic stuff is to take a look at what is in the store when handling a ticket.
The priorities play an Important Role while handling huge volumes of tickets and when there is some SLA rules being applied to each incoming tickets. There are generally tagged as Low , Normal, Medium, Critical and Urgent.
According to the ITIL (Information Technology Infrastructure Library) rules and concept, the priority of a ticket is driven by two factors - Urgency and Impact.
Impact refers to the contents of incoming ticket which will affect downtime to numerous customers or affect the company as such. For eg : A ticket which describes a security vulnerability in one of the application may impact a network downtime and there by the business. Now this will be considered as a Critical ticket.
Urgency refers to those incoming tickets which needs to be handled as soon as possible. Lets say a ticket is opened by one of the high paying customers and he is eagerly waiting for the Support's response. This will be treated as a Critical ticket.
So basically, the priority of the ticket not just applies to the status the ticket shows, but also should be handled according to the discretion of the agent considering the gravity of the situation.