The definitions of support impact are in the Statment of Support.
If a customer submits a ticket via the web ticket form, they can choose the starting priority of the ticket - this is the Customer Priority field you will see in Zendesk. On ticket creation a trigger sets the main Priority field (not visible to the customer) to match the Customer Priority choice. After ticket creation, changes to Customer Priority do not affect the Priority field or SLAs.
If a customer emails in a ticket it will get a Priority of 'Normal' (unless it is sent to the special emergency contact).
Manually setting a ticket's priority in the Priority field (not Customer Priority) in Zendesk will change the overall ticket SLA, for both the first and next replies. This allows support to prioritize tickets and update the urgency during the life of the ticket (for example the initial request may be 'High' priority and then follow up questions may need 'Low' priority.) Customers don't receive automatic notifications upon priority change.
Everyone is responsible for setting priority during ticket triage. If it comes to pass that a ticket has an incorrect priority, feel free to reset the priority.
There is a Zendesk Macro General::Changed priority that includes language and links that will be helpful if you need to do this.
Note that the amount of time that passes between our last response and the next response from the customer does not impact the priority of the ticket. For example, if the ticket should have High priority based on the definitions of support impact and the customer only responds to our initial query after a month, the ticket priority should still remain High (assuming the nature of the problem did not change).
Changing the priority in a ticket will change the SLA, because different priorities have shorter or longer SLAs. Please refer to the guidelines and SLA listed in our statement of support.